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A  TREATISE 


ox 


BOX  OF   INSTRUMENTS 


AXD 


THE   SLIDE-RULE. 


FOR  THE  USE  OF  GACGERS,  ENGINEERS,  SEAMEN, 
AND  STUDENTS. 


BY    THOMAS    KENTISH. 


PHILADELPHIA  : 
HEXRY    CAREY  -BATED, 

INDUSTRIAL    PUBLISHER. 

No.    406    WALNUT    STREET. 

1872. 


PREFACE. 


IN  the  first  edition  of  this  Treatise,  its  utility,  in 
the  absence  of  other  works  upon  the  subject,  was 
assigned  as  an  apology  for  its  publication.  The  in- 
struments, whose  uses  it  explains,  are  often  so  little 
understood,  that  scarcely  half  of  'them  are  of  any 
service  to  their  possessor.  The  Sector,  in  particular, 
the  most  important  in  the  box,  is  generally  regarded 
as  unintelligible.  The  Slide-rule  is  briefly  noticed 
in  some  of  the  treatises  on  Mensuration ;  but,  as  the 
pupil  is  presented  merely  with  a  few  formal  precepts 
how  to  use  it,  without  knoAving  why,  he  never  under- 
stands its  nature,  never  understands  the  method  of 
determining  the  real  value  of  any  result,  and,  ac- 
cordingly, soon  lays  it  by  with  dissatisfaction,  and 
banishes  it  from  his  memory. 

The  steady  sale  which  the  first  edition  has  met 
with  has  convinced  the  Author  that  his  labours  were 
not  in  vain,  and  that  he  has  extended  among  many 
thousands  a  knowledge  of  intrinsic  value  to  all  em- 
ployed in  the  delineation  of  mathematical  figures. 

3 


4  PREFACE. 

No  attempts,  however,  are  perfect  in  the  beginning ; 
and  much  was  wanted  in  the  first  edition  to  render 
the  work  complete.  This  additional  information 
has  been  supplied.  Several  problems  have  been 
prefixed,  requiring  only  the  compasses  and  ruler, 
which,  together  with  those  that  follow,  embrace  all 
that  are  truly  useful,  and  preclude  the  necessity  of 
referring  to  other  works  on  Practical  Geometry. 

In  books  upon  this  subject,  it  is  not  usual  to  annex 
reasons  for  any  of  the  operations,  but  it  has  been 
thought  advisable  to  do  so,  in  a  few  instances,  with 
the  more  difficult  problems ;  with  the  rest  it  is  not 
attempted,  because,  to  have  entered  fully  upon  the 
subject,  would  have  been  to  transcribe  the  whole  of 
the  Elements  of  Euclid,  a  work  which  is  within  the 
reach  of  every  one,  and  which  every  one  must  study, 
who  desires  thoroughly  to  understand  Geometry. 

The  part  relating  to  Trigonometry,  though  concise, 
will  be  found  to  comprehend  every  thing  necessary 
to  enable  the  student  to  obtain  a  clear  conception 
of  the  subject,  and  when  carried  out  in  connection 
with  the  portion  devoted  to  Navigation,  will  render 
its  acquirement  alike  easy,  pleasing,  and  useful. 

The  chapter  on  Logarithms  is  written  simply  to 
show  the  mode  of  adapting  them  to  instrumental 
computation ;  a  purpose  to  which  every  part  of  tho 
work  is,  as  a  matter  of  course,  as  much  as  possible 
made  subservient. 

The  section  relating  to  the  Slide-rule  has  been 
entirely  re- written ;  and,  in  this  portion  of  the  work, 
the  Author  flatters  himself  there  will  be  found  much 


PKEFACE.  5 

that  is  perfectly  new,  and  many  remarks  calculated 
to  awaken  and  stimulate  the  youthful  mind  to  think 
for  itself;  a  habit  of  the  utmost  value  in  mathema- 
tical science,  which,  being  based  on  truth,  courts  in- 
vestigation, and  requires  that  we  shall  never  assent 
till  we  can  comprehend.  In  this  part,  the  formulae 
for  surfaces  and  solids  have  been  so  modified  as  to 
embrace  almost  every  species  of  mensuration  under 
the  simplest  form ;  questions  for  practice  are  inter- 
spersed throughout,  that  the  student  may  test  his 
proficiency,  and  acquire  facility  in  the  use  of  the 
rule;  and  tables  are  inserted  at  every  step,  for  the 
purposes  of  computation ;  a  practice  in  all  cases  ad- 
visable, as  the  instrumental  operation  and  numerical 
calculation  necessarily  check  and  illustrate  each 
other. 

The  reciprocals  of  divisors,  employed  as  factors, 
are  convenient  in  practice;  but  it  was  deemed  ad- 
visable, upon  the  whole,  to  omit  them,  as  the  formulae 
for  numerical  computation  would  have  then  been 
different  from  those  suited  to  the  Slide-rule,  which 
would  have  tended  to  perplex  the  mind  of  the 
learner ;  whereas,  by  retaining  the  same  form  for 
both  operations,  it  is  obvious  that  to  understand  one 
is  to  understand  the  other ;  and  the  student,  instead 
of  coming  to  regard  the  instrumental  mode  of  solu- 
tion as  something  entirely  distinct  from  the  nume- 
rical, and  looking  upon  the  agreement  of  the  two 
rather  as  a  coincidence  than  a  consequence,  as  is 
too  often  the  case,  will  see  that,  in  fact,  they  are 
identical,  and  'cannot  fail,  in  a  short  time,  of  having 


i* 


6  PREFACE. 

the  very  clearest  conception  of  the  whole  of  the 
subjects  treated  of. 

It  is  somewhat  surprising,  that,  after  the  lapse  of 
two  hundred  years,  so  excellent  an  instrument  as  the 
Slide-rule  should  he  so  little  known  and  appreciated 
by  mathematical  students  in  general.  To  the  engi- 
neer and  the  excise  officer  it  is  perfectly  familiar, 
and  of  daily  utility;  but,  from  its  having  been  al- 
most exclusively  confined  to  them,  there  is  an  idea 
prevalent  among  gentlemen  engaged  in  education, 
that  it  can  neither  be  understood  by  their  pupils 
nor  be  of  any  utility  to  them.  A  more  erroneous 
conception,  on  both  accounts,  cannot  be  formed; 
for  a  knowledge  of  the  instrument  is  acquired  with 
little  or  no  effort.,  and  it  may  be  truly  stated,  that 
it  is  the  most  valuable  adjunct  to  mathematical 
study  that  can  possibly  be  desired.  Nothing  im- 
prints a  fact  so  firmly  on  the  mind  as  repeated 
exercise.  As  Demosthenes,  when  asked  the  three 
principal  requisites  in  oratory,  summed  them  up  in 
the  word  action ;  so  may  we  say  of  learning,  that 
the  three  great  essentials  to  its  success  are  contained 
in  the  word  repetition.  Dexterity  in  every  art,  and 
skill  in  every  science,  must  be  acquired  by  this 
means,  and  by  this  alone.  But,  in  the  solution  of 
questions  that  are  necessarily  laborious,  every  one 
feels  a  great  disinclination  to  work  through  many 
examples,  much  less  to  repeat  them ;  the  consequence 
is,  so  little  impression  is  made  on  the  memory,  that 
the  knowledge  is,  in  many  instances,  forgotten  as 
soon  as  acquired.  Now,  by  the  Slide-rule,  the  most 


PREFACE.  7 

tedious  calculations  are  effected  nearly  as  easily  as 
the  most  simple.  The  student  can,  therefore,  after 
accompanying  them  the  first  time  with  the  numerical 
solution,  go  over  the  operations  again  and  again 
with  the  rule,  with  the  greatest  ease  and  rapidity, 
deepening  the  impression  each  succeeding  time,  and 
rendering  the  knowledge  obtained  distinct  and  per- 
manent. 

In  the  truth  of  this,  the  Author  is  not  only  borne 
out  by  his  own  experience,  but  he  can  refer,  with 
pleasure,  to  schools  in  which  they  have  been  adopted, 
and  in  which  they  have  proved  of  the  greatest  assist- 
ance; and  no  one,  really  fond  of  knowledge,  who 
may  give  them  a  trial,  will  regret  the  little  extra 
trouble  they  may  cause,  but  will  rejoice  in  having 
found  so  excellent  an  aid  to  study.  Mathematical 
science  is  of  such  extensive  utility  that  it  ought  to 
be  universally  understood ;  and  it  is  impossible  to 
go  five  or  six  times  through  the  present  work,  which, 
after  the  first,  may  be  done  in  a  very  few  days,  with- 
out being  as  familiar  with  the  Surfaces  and  Solids, 
and  with  Trigonometry  and  Navigation,  as  with  the 
multiplication  table;  and  this  is  the  great  object  to 
be  attained.  To  be  barely  acquainted  with  them  is 
not  sufficient ;  knowledge,  to  be  useful,  must  be  at 
the  moment  accessible,  so  that  we  may  be  enabled 
to  proceed  without  error  or  hesitation ;  and  that  the 
most  intimate  familiarity  with  the  above-mentioned 
studies  will  be  obtained  by  the  method  here  pointed 
out,  has  been  again  and  again  tried,  and  with  the 
happiest  results.  v 


8  PREFACE. 

The  small  section  allotted  to  Land  Surveying  doea 
not  properly  come  within  the  design  of  the  work, 
but  it  was  thought  it  might  prove  useful,  and  has  ac- 
cordingly been  inserted.  The  measuring  of  a  field, 
which  is  all  that  can  at  all  be  consistently  aimed  at 
here,  is  so  very  simple,  that  one  example  was  deemed 
sufficient  as  a  guide;  but,  in  teaching  the  subject, 
more  is  necessary;  and  a  very  efficient  method  is  to 
draw  on  a  piece  of  paper  a  sketch  of  a  field,  which, 
with  the  help  of  a  feather-edged  plotting  scale,  or  a 
diagonal  scale  and  a  pair  of  compasses,  the  pupil 
should  measure,  and  enter  his  notes  in  a  field-book, 
or  slate,  ruled  for  the  purpose.  The  sketch  should 
now  be  handed  to  the  tutor.  The  learner,  then, 
from  his  notes,  is  to  construct  another,  upon  paper, 
from  the  same  scale.  When  finished,  its  correctness 
can  be  readily  ascertained  by  laying  it  upon  the 
original,  and  holding  them  up  to  the  light,  when,  if 
accurately  laid  down,  the  lines  will,  of  course,  cor- 
respond. This  plan  has  been  tried  for  many  years, 
and  found  to  convey  a  very  good  idea  to  the  mind 
of  the  learner.  A  little  occasional  field-practice, 
which  is  indispensably  requisite,  soon  renders  the 
study  pleasant,  and  the  progress  certain. 

The  chapter  on  Cask  Gauging  will,  it  is  humbly 
hoped,  prove  a  valuable  acquisition  to  the  gauger. 
The  great  uncertainty  and  inconvenience  of  the  four 
varieties  render  it  extremely  desirable  to  have  some 
general,  and,  at  the  same  time,  easy,  and  easily  re- 
membered rule  of  approximation;  and  from  the 
method  employed  in  making  casks,  it  is  obvious  that 


PREFACE.  9 

the  exact  agreement  of  their  shape  with  any  definite 
geometrical  solid  must  be  perfectly  fortuitous.  The 
four  varieties,  however,  are  exemplified,  together 
with  the  general  rule  for  frustums;  which  latter, 
though  rather  tedious,  would  soon  become  familiar 
if  once  adopted. 

In  Navigation,  for  working  a  day's  reckoning,  the 
rule  is  peculiarly  convenient,  and  sufficient  for  all 
practical  purposes;  superseding  the  incessant  turn- 
ing over  and  transcribing  from  tables ;  which,  though 
in  themselves  they  are  one  of  the  most  splendid  in- 
ventions of  all  time,  and,  in  elaborate  calculations 
requiring  minute  exactness,  indispensable,  are  yet, 
in  their  application,  as  perfectly  mechanical  as  the 
instrumental  operation  itself;  so  that  no  reasonable 
objection  can  be  urged  against  the  adoption  of  the 
Gunter,  that  does  not  apply,  with  equal  force,  to 
the  use  of  Logarithms  altogether. 

For  gentlemen,  however,  who  may  not  desire  to 
use  the  Slide-Rule,  it  may  be  here  stated,  that  the 
work  by  no  means  absolutely  requires  it ;  it  is  equally 
available  as  a  Treatise  on  Mensuration,  Trigonome- 
try, and  Navigation.  For  the  purposes  of  calcula- 
tion, it  would  be  found  a  great  convenience  to  copy 
out,  upon  a  sheet  of  Bristol  board,  the  tables  at 
pages  115,  116,  118, 123,  126,  129,  136,  137,  138, 
150,  182,  and  198,  as  it  would  save  much  needless 
turning  over  of  the  pages ;  and  if  each  were  enclosed 
in  borders,  and  slightly  washed  over  with  different 
colours,  it  would  make  them  of  easier  reference. 

In  studying  Trigonometi  y,  Wallace's  Practical 


10  PREFACE. 

Mathematician's  Pocket  Guide  will  be  found  a  con- 
venient set  of  Logarithmic  Tables ;  their  cost  is  a 
mere  trifle.  Barlow's  and  Galbraith's  Tables  are 
extremely  useful.  The  latter  contains  the  secants 
and  cosecants,  which,  as  complemental  to  the  cosines 
and  sines,  offer  great  facilities  in  calculation. 

Kentish's  Slide-Rules,  furnished  with  triple  slides, 
are  sold  by  Messrs.  Relfe  Brothers,  150  Aldergate 
Street.  London  price,  carefully  mounted  and  var- 
nished, 5s.  each,  or  beautifully  cut  in  box,  10s.  each. 
Messrs.  J.  "W.  Queen  &  Co.,  Importers,  Manufactu- 
rers and  Dealers  in  Mathematical  Instruments,  924 
Chestnut  Street,  Philadelphia,  will  import  them  to 
order. 


CONTENTS. 


A  Box  OP  ISSTBUXXSTS 13 

PRACTICAL  GEOMETRY — 

Definitions 15 

The  Compasses 20 

The  Parallel  Ruler 27 

The  Protractor 36 

The  Plain  Scale 40 

TRIGONOMETRY 48 

The  Sector  54 

LOGARITHM? 81 

The  Slide-role 101 

Ratios  and  Gauge  Points Ill 

TABLE  I.  The  Circle 112 

II.  Polygons,  linear  dimensions  of 115 

IIL  Polygons,  areas  of 117 

IV.  Falling  Bodies 119 

Pendulums 120 

Areas  of  Circles  and  Surfaces  of  Spheres....  120 

Diagonals  of  Squares  and  Cubes. 122 

Velocity  of  Sound 123 

V.  Surfaces 124 

Formulae  for  ditto 126 

11 


12  CONTENTS. 

PAGl 

TABLE   VI.  Solids 131 

General  Rule  for  Frustums 133 

Formulae  for  Solids 136 

VII.  Weight  of  Spheres 154 

VIII.  Solidity  of  Spheres 155 

Solar  System 156 

Miscellaneous  Questions 158 

Cask  Gauging 163 

Land  Surveying 177 

TEIGOSOMETBT  ASD  NAVIGATION 180 

Plane  Sailing 183 

Traverse  sailing 186 

Parallel  Sailing 188 

Middle  Latitude  Sailing 191 

Mercator's  Sailing 197 

Oblique  Sailing 204 

Windward  Sailing 206 

Current  Sailing 209 

Of  a  Ship's  Journal 211 

RECAPITULATION 216 

APPENDIX , J23 


A  TREATISE 


A  BOX  OF  INSTRUMENTS 


A  BOX  OF  INSTRUMENTS. 

THE  contents  of  a  case  of  Mathematical  Instruments 
are,  generally,  a  pair  of  plain  compasses,  a  pair  of  bow 
compasses,  a  pair  of  drawing  compasses,  and  a  drawing 
pen ;  a  parallel  ruler,  a  protractor,  a  plain  scale,  and  a 
sector.  The  plain  compasses  consist  of  two  inflexible  rods 
of  brass,  revolving  upon  an  axis  at  the  vertex,  and  fur- 
nished with  steel  points.  The  bow  compasses  are  a 
smaller  pair,  provided  with  a  pen  for  describing  small 
circles  in  ink.  The  sides  of  the  pen  are  opened  or  closed 
with  an  adjusting  screw,  that  the  line  may  be  drawn  fine 
or  coarse  as  required.  The  drawing  compasses  are  the 
largest  of  the  three ;  one  of  the  legs  is  furnished  with  a 
socket  for  the  reception  of  either  of  the  four  following 
pieces,  as  occasion  may  require  : — 1.  A  steel  point,  which, 
being  fixed  in  the  socket,  makes  the  compasses  a  plain 
pair,  like  the  other ;  2.  A  port-crayon,  for  the  purpose  of 
carrying  a  piece  of  blacklead,  or  slate-pencil,  according  as 
paper  or  slate  is  used  for  drawing  upon ;  3.  A  steel  pen, 
like  the  one  attached  to  the  bow  compasses,  but  larger,  for 
the  purpose  of  describing  circles  of  greater  diameter ;  4.  A 


14  A   TREATISE   ON   A   BOX   OF 

rowel,  or  spur  wheel,  with  a  brass  pen  above  it,  for  the  re- 
ception of  ink,  which  the  spur,  in  its  circuit,  distributes  in 
dots  upon  the  paper.  The  pen,  the  port-crayon,  and  the 
dotting  wheel,  are  each  furnished  with  a  joint,  that,  when 
fixed  in  the  compasses,  they  may  be  set  perpendicular  to 
the  paper.  The  drawing  pen  is  the  same  as  the  steel  pen 
of  the  compasses,  only  that  it  is  screwed  upon  a  brass  rod, 
of  a  convenient  length  for  the  hand,  and  into  the  rod 
itself  is  inserted  a  fine  steel  point  for  pricking. 

The  parallel  ruler  consists  of  two  flat  pieces  of  ebony  or 
ivory,  connected  together  by  brass  bars,  having  their  ex- 
tremities equidistant,  by  which  contrivance,  when  the 
ruler  is  opened,  the  sides  necessarily  move  in  parallel 
lines.  The  protractor  is  a  semicircular  piece  of  brass,  di- 
vided into  180  degrees,  and  numbered  each  way,  from 
end  to  end.  In  some  boxes  this  is  omitted,  and  the  de- 
grees are  transferred  to  the  border  of  the  plain  scale.  The 
plain  scale  is  a  flat  piece  of  box  or  ivory,  and  is  so  called 
from  containing  a  number  of  lines  divided  into  plain  or 
equal  parts.  A  scale  of  chords,  of  a  fixed  radius,  is  also 
graduated  upon  it.  The  sector  is  a  foot  rule,  divided  into 
equal  portions,  movable  upon  a  brass  joint,  or  axis,  from 
the  centre  of  which  are  drawn  various  lines  through  the 
whole  length  of  the  ruler.  The  legs  represent  the  radii 
of  a  circle,  and  the  middle  of  the  joint  expresses  the 
centre.  The  lines  upon  the  sector  are  of  two  sorts,  single 
and  double :  the  single  lines  run  along  the  margin  and 
the  edges ;  the  double  lines  radiate  from  the  centre  to  the 
extremities  of  the  legs,  and  are  marked  twice  upon  the 
same  face  of  the  instrument,  in  order  that  distances  may 
be  taken  upon  them  crosswise,  when  they  arc  opened  to 
an  angular  position. 


INSTRUMENTS    AND   THE    SLIDE-RULE. 


15 


PRACTICAL  GEOMETRY. 


DEFINITIONS. 

A  POINT  is  that  which  has  position,  without  length, 
breadth,  or  thickness. 

A  line  is  length,  without  breadth,  or  thickness. 

A  superficies  is  length  and  breadth,  without  thickness. 

A  solid  is  that  which  has  length,  breadth,  and  thick- 
ness. 

An  angle  is  the  opening  of  two  straight  lines  meeting 
in  a  point,  as  RAE. 

Lines  which  run  side  by  side,  and  are  always  equidis- 
tant, are  called  parallels,  as  SD,  KL. 

A  line  is  perpendicular  to  another  when  the  angles  on 
both  sides  of  it  are  equal  j  and  each  of  these  angles  i? 


36  A   TREATISE   ON   A   BOX   OF 

called  a  right  angle ;  thus  RA  is  perpendicular  to  WS  f 
and  the  angles  RAW,  RAS,  are  right  angles. 

An  acute  angle  is  less  than  a  right  angle,  as  EAS. 

An  obtuse  angle  is  greater  than  a  right  angle,  as  TAS 

A  figure  containing  three  sides  is  called  a  triangle. 

The  boundary  of  a  right-lined  figure  is  termed  its 
perimeter. 

A  triangular  figure  containing  three  equal  sides  is  an 
equilateral  triangle,  as  KBC. 

If  two  of  its  sides  only  are  equal,  it  is  an  isosceles 
triangle,  as  BHC,  in  which  BH  =  HC. 

If  the  three  sides  are  unequal,  it  is  a  scalene  triangle, 
as  KBH. 

A  triangle  containing  a  right  angle  is  called  a  right- 
angled  triangle,  as  ABC. 

A  triangle  containing  an  obtuse  angle  is  an  obtuse- 
angled  triangle,  as  KHB. 

An  acute-angled  triangle  contains  three  acute  angles, 
asNMC. 

A  figure  containing  four  sides  is  called  a  quadrilateral. 

A  parallelogram  is  a  quadrilateral  whose  opposite  sides 
are  parallel,  as  SKLD,  KPOL. 

A  rectangle  is  a  parallelogram  whose  angles  are  righ* 
angles,  as  KMCL. 

A  square  is  a  rectangle  whose  sides  are  equal,  as  ABCD. 

A  rhomboid  has  its  opposite  sides  equal,  but  two  of  its 
angles  are  obtuse,  and  two  acute,  and  these  are  opposit- 
to  each  other,  and  equal,  each  to  each,  as  KBML. 

A  rhombus,  like  a  square,  has  all  its  sides  equal,  but 
two  of  its  angles  are  obtuse,  and  two  acute,  and  these  are 
opposite  to  each  other,  and  equal,  each  to  each,  ay  PBMO. 


INSTRUMENTS    AND   THE    SLIDE-RULE.  17 

When  a  quadrilateral  has  only  one  pair  of  its  sides 
parallel,  it  is  called  a  trapezoid,  as  PBCO. 

When  none  of  its  sides  are  parallel  it  is  a  trapezium, 
as  KHNL. 

A  line  crossing  a  quadrilateral  from  opposite  angles,  is 
termed  a  diagonal ;  thus  AC,  BD,  are  the  diagonals  of  the 
square  ABCD. 

Figures  of  more  sides  than  four  are  called  polygons. 

If  all  the  sides  and  angles  are  equal,  it  is  a  regular 
polygon ;  if  unequal,  an  irregular  polygon. 

A  polygon  of  five  sides  is  termed  a  pentagon ;  of  six,  a 
hexagon ;  of  seven,  a  heptagon  ;  of  eight,  an  octagon ;  of 
nine,  a  nonagon ;  of  ten,  a  decagon ;  of  eleven,  an  un- 
decagon  ;  and  of  twelve,  a  dodecagon. 

A  triangle  is  sometimes  called  a  trigon  j  and  a  quadri- 
lateral, a  tetragon. 

A  circle  is  a  plane  figure  bounded  by  a  curved  line 
called  the  circumference,  which  is  everywhere  equidistant 
from  the  centre. 

A  right  line  passing  through  the  centre,  and  meeting 
the  circumference  at  each  extremity,  is  called  the  diameter, 
asSM. 

A  right  line  reaching  from  the  centre  to  the  circumfe- 
rence is  termed  the  radius,  as  HM. 

An  arc  of  a  circle  is  any  part  of  the  circumference,  as 
the  curve  from  S  to  X. 

A  chord  is  a  right  line  joining  the  extremities  of  an 
arc,  as  the  straight  line  from  S  to  X. 

A  segment  is  a  space  included  between  an  arc  and  its 
chord,  as  SZXS. 

A  sector  is  a  part  of  a  circle  contained  by  two  radii  and 
*he  arc  between  them,  as  SHXZ. 


18  A  TREATISE   ON   A   BOX   OF 

Hence  a  sector  is  made  up  of  a  triangle  and  a  segment 

A  semicircle  is  half  a  c.rcle;  a  quadrant,  the  fourth 
part;  a  sextant,  the  sixth  part;  and  an  octant,  the  eighth 
part. 

A  rectilineal  solid  whose  ends  are  equal,  similar,  and 
parallel,  and  whose  sides  are  parallelograms,  is  called  a 
prism. 

If  the  ends  also  of  the  prism  are  parallelograms,  it  is  a 
parallelepiped  :  if  all  the  sides  are  square,  it  is  a  cube  :  if 
the  ends  are  unequal  and  dissimilar,  it  is  a  prismoid. 

A  cylinder  is  a  round  solid,  of  uniform  thickness,  hav- 
ing circular  ends. 

A  pyramid  is  a  solid  which  has  a  rectilineal  base,  and 
triangular  sides  meeting  in  a  point  called  the  vertex. 

A  cone  is  a  round  solid  tapering  uniformly  to  a  point. 

A  sphere  is  a  solid  every  way  round. 

A  segment  of  a  solid  is  the  part  cut  off  the  top  by  a 
plane  parallel  to  its  base. 

A  frustum  is  the  part  left  at  the  bottom,  after  the  seg- 
ment has  been  cut  off. 

Prisms,  cylinders,  pyramids,  and  cones  are  said  to  be 
right  or  oblique  according  as  the  base  is  cut  perpendicu- 
larly or  obliquely  to  the  axis. 

Plain  figures  formed  by  the  cutting  of  a  cone  are  called 
conic  sections.  A  cone  may  be  cut  five  ways.  If  the 
cutting  plane  passes  through  the  vertex  of  a  right  cone 
and  any  part  of  the  -base,  the  section  is  an  isosceles  tri- 
angle ;  if  through  the  sides,  parallel  to  the  base,  a  circle  ; 
if  obliquely  through  the  sides,  an  ellipse ;  if  through  one 
side  and  parallel  to  the  other,  a  parabola ;  if  in  any  other 
way,  the  cutting  plane  will  run  beyond  into  a  similar  cone 
inverted  over  the  other,  and  then  the  section  is  termed  ac 


LNslRUMEXTS    AND   THE    SLIDE-RULE.  19 

hjperbola.  (JVI  B.  An  ellipse  may  be  considered  iu  an 
elongated  circle,  and  may  be  described  by  driving  in  tiro 
pins  as  centres,  passing  over  them  a  string  with  a  loop  at 
each  end,  and  working  a  pencil  round  within  the  string, 
keeping  it  stretched  to  its  limits.)  The  centres  are  usually 
called  foci :  the  diameter  passing  through  them  is  termed 
the  transverse  ;  the  short  one  at  right  angles  to  it,  the  con- 
jugate diameter. 

A  line  perpendicular  to  either  of  the  diameters  is  called 
an  ordinate  ;  and  the  sections  of  the  diameter  met  by  it 
are  termed  abscissas. 

The  vertex  of  a  conic  section  is  the  point  where  the 
cutting  plane  meets  the  opposite  sides  of  the  cone.  The 
axis  of  a  parabola  or  hyperbola  is  a  right  line  drawn  from 
the  vertex  to  the  middle  of  the  base. 

All  round  solids  may  be  conceived  to  be  described  by 
the  rotation  of  planes  on  their  sides,  or  diameters,  as  axes. 

A  right-angled  triangle  rotating  on  its  perpendicular, 
forms  a  cone ;  a  parallelogram,  revolving  on  its  side,  pro- 
duces a  cylinder ;  if  a  circle  turn  upon  its  diameter,  it 
shapes  out  a  sphere  ;  and  the  revolution  of  an  ellipse  gene- 
rates a  spheroid.  If  the  ellipse  revolves  on  the  transverse 
diameter,  the  spheroid  is  called  prolate ;  if  on  the  conju- 
gate, an  oblate  spheroid.  The  figure  formed  by  the  revo- 
lution of  a  parabola  about  its  axis  is  termed  a  paraboloid, 
or  parabolic  conoid ;  the  solid  formed  in  the  same  way 
by  an  hyperbola,  an  hyperboloid,  or  hyperbolic  concoid. 

If  a  section  of  a  curve  revolve  on  a  double  ordinate  as 
axis,  it  will  generate  a  spindle ;  and  this  will  be  circular, 
elliptic,  parabolic,  or  hyperbolic,  according  to  the  curve 
from  which  the  section  is  taken. 

A  regular  body  id  a  solid  contained  under  a  certain 


20 


A   TREATISE   ON   A  BOX   OP 


number  of  similar  and  equal  plane  figures.  There  are  but 
five  kinds,  which  are,  the  tetraedron,  Laving  four  equal 
triangular  faces ;  the  hexaedron,  or  cube,  which  has  six 
equal  square  faces ;  the  octaedron,  which  has  eight  equal 
triangular  faces ;  the  dodecaedron,  which  has  twelve  equal 
pentagonal  faces;  and  the  icosaediwn,  which  has  twenty 
equal  triangular  faces. 


THE  COMPASSES 
1.  To  bisect  a  given  line  AB. 


From  A  and  B  as  centres,  with  any  radius  greater  than 

half  AB,  describe  arcs  cutting  each  other  in  C  and  D. 

Join  the  points  C  and  D,  by  drawing  the  straight  line  CD; 

this  will  be  perpendicular  to  AB,  which  it  will  bisect  in 

the  point  E. 

2.  To  bisect  a  given  angle  ABC.    (See  p.  21.) 

From  B  as  a  centre,  with  any  radius,  describe  the  arc 

DE.     From  D  and  E,  with  the  same,  or  any  other  radius, 

draw  arcs  cutting  each  other  in  F.     Join  BF,  and  it  will 

bisect  the  angle  as  required. 


INSTRUMENTS   AND   THE    SLIDE-RULE.  21 


3.  To  erect  a  perpendicular  to  a  straight  line  AB,  from 
a  given  point  C  within  it. 


DOE 

When  the  point  C  is  near  the  middle  of  the  line,  on 
each  side  of  it  set  off  any  two  equal  distances  CD,  CE. 
From  D  and  E  as  centres,  with  any  radius  greater  than 
CE  or  CD,  describe  arcs  cutting  each  other  in  F.  Join 
FC,  and  it  will  be  perpendicular  to  AB. 


D  C 

When  the  point  C  is  at  or  near  the  end  of  the  line,  from 
C,  with  any  radius,  describe  the  <rrs  DBF.     From  D,  with 


22 


A   TREATISE   ON   A   BOX   OF 


the  same  radius,  cross  it  in  E.  From  E,  with  the  same 
radius,  describe  the  arc  GF ;  and  from  F,  with  the  same 
radius,  cross  the  last  arc  at  G.  Lraw  GO,  and  it  will  be 
perp  ndicular  to  AB. 

4.  To  draw  a  perpendicular  to  a  line  AB,  from  a  point 
C  wit  on*  it 

C 


When  the  p*>j?l  \!  i  QI  t»ily  opposite  the  middle  ol  the 
line,  from  C,  wili«  wy  sonvenient  radius,  describe  the  arc 
DFE  crossing  Aft  in  X>  tod  E.  From  D  and  E,  with 
the  same  or  any  t  tf  •.*  radius,  describe  arcs  cutting  each 
other  in  G.  Draw  0  '.I,  uid  it  will  be  perpendicular  to  Alt 


When  the  point  C  is  nearly  opposite  the  end  of  the  line, 
from  C  draw  any  line  CD.  Bisect  CD  in  E,  and  from 
E,  with  the  radius  EC,  cross  AB  in  F.  Draw  CF,  and 
it  will  h*>  wpoMioi^ar  to  AT* 


INSTRUMENTS  AND   THE   SLIDE-RULE.  23 

5.  To  make  an  angle  equal  to  a  given  angle  ABO. 


From  B,  with  any  radius,  draw  the  arc  GH;  and  from 
E,  with  the  same  radius,  describe  the  arc  KL.  Make 
KL  equal  to  GH,  and  through  K  draw  the  straight  line 
ED.  The  L  DBF  =  L  ABC. 

6.  To  describe  a  circle  through  three  given  points  A.,  B, 
an4C. 


From  B,  with  any  radius  less  than  BA,  describe  the  aro 
alcd;  and  from  A  and  C,  with  the  same  radius,  cross  it 
in  a  and  I,  c  and  d.  Draw  straight  lines  through  the 
points  of  intersection  to  moot  in  D,  which  will  be  the 
centre  of  the  circb  required. 


24  A   TREATISE   ON   A   BOX   OF 

To  find  the  centre  of  a  given  circle,  take  any  three 
points  in  the  circumference,  and  proceed  in  like  manner. 

To  describe  a  circle  about  a  triangle,  select  the  three 
angular  points,  and  proceed  in  like  manner. 

7.  To  construct  a  triangle  of  which  the  three  sides  A, 
B,  and  C  are  given. 


Draw  the  line  DE  equal  to  A.  From  D,  with  0  for  <» 
radius,  describe  an  arc  at  F ;  and  from  E,  with  B  for  a 
radius,  cross  it  at  F.  Draw  the  lines  DF,  EF;  and  DEF 
will  be  the  triangle  required. 

To  construct  an  equilateral  triangle  proceed  in  the  same 
manner,  taking  the  base  each  time  as  radius. 

8.  To  construct  a  rectangle,  whose  length  A  B  and 
breadth  C  are  giveli. 


At  A  erect  a  perron  Ocular  41),  equal  to  C.     From  \)t 


INSTRUMENTS   AND  THE    SLIDE-RULE.  25 

with  the  distance  AB,  describe  an  arc  at  E  ;  and  from  B, 
with  the  radius  C,  cross  it  at  E.  Draw  DE,  EB. 
A  DEB  is  the  parallelogram  required. 

To  construct  a  square,  make  the  perpendicular  equal  to 
the  base,  and  proceed  in  like  manner. 

To  construct  a  rhombus  and  rhomboid,  determine  the 
angle  by  problem  5,  and  then  proceed  in  like  manner. 

9.  To  draw  a  line  parallel  to  a  given  line  AB,  at  a 
given  distance. 

G  H 


Take  any  two  points  E  and  F  in  the  line  AB,  and,  with 
the  given  distance,  describe  the  arcs  G  and  H.  Draw  the 
line  CD  touching  the  arcs,  and  it  will  be  parallel  to  AB. 

When  the  line  is  to  pass  through  a  given  point  C. 


In  AB  take  any  point  G,  and  with  the  distance  GO 
describe  the  arc  CH  ;  from  C,  with  the  same  radius,  de- 
scribe the  arc  GF,  and  make  FG  equal  to  CH.  Through 
F  and  C  draw  the  straight  line  DE,  and  it  will  be  paral- 
lel to  AB,  as  required. 


26 


A   TREATISE   ON   A    BOX   OF 


10.  To  project  an  ellipse,  or  oval,  the  length  A  B  and 
breadth  A  C,  being  given. 


Bisect  AC  in  D,  and  upon  it  describe  the  semicircle 
AEG.  At  C  draw  a  straight  line  CB,  perpendicular  to 
AC ;  and  from  the  point  A,  with  the  given  length  as  a 
radius,  cross  CB  in  B.  From  the  semicircle  AEC  and 
parallel  to  CB,  draw  any  number  of  straight  lines  FGr<7, 
HK&,  NMm,  EDrf,  &c.  On  the  line  AB,  at  the  points 
of  intersection,  m,  Te,  g,  &c.  erect  perpendiculars,  and  make 
gf  equal  to  GF,  Teh  equal  to  KH,  mn  equal  to  MN,  &c. 
Lastly,  trace  a  curve  line  from  B  through  the  points  /,  h, 
n,e,  &c.,  and  it  will  give  half  of  the  ellipse,  from  which 
the  other  half  may  readily  be  constructed.  This  method 
is  of  great  utility  in  describing  elliptical  arches,  stair- 
cases, &c.,  and  for  any  purpose  in  which  circular  figures, 
or  figures  of  any  shape,  require  to  be  elongated  without 
altering  the  breadth,  as  in  cutting  gores  for  globes,  &c., 


INSTRUMENTS   AND   THE   SLIDE-RULE.  27 

in  which  case  AB  will  be  equal  to  half  the  circumference, 
the  breadth  DE  being  regulated  by  the  number  of  gores 
employed.  la  instances  like  these,  the  length  AB  will 
he  very  great  compared  with  the  breadth  DE,  and  AB 
may  then  be  divided  in  the  same  proportion  as  AC,  by 
other  means ;  for  example,  if  the  length  is  to  be  12  times 
the  breadth,  then  each  of  the  distances  B^r,  gk,  Jem,  will 
be  12  times  the  corresponding  distances  CG-,  GK,  KM. 
This  method  of  projecting  ellipses  is  derived  from  conceiv- 
ing a  right  cylinder  to  be  cut  by  two  planes,  one  parallel, 
and  the  other  oblique  to  the  base. 

THE  PARALLEL  RULER. 

1.  Through  a  given  point  A,  to  draw  a  line  parallel  to 
a  given  line  DE. 


Lay  the  edge  of  the  ruler  upon  DE,  and  move  it  up- 
wards  till  it  reaches  the  point  A,  through  which  draw  BG. 
BC  is  parallel  to  DE. 

2.  To  make  an  angle  equal  to  a  given  angle  ABC. 
A 


Lay  the  base  EF,  in  a  line  with  BC,  and   draw  ED, 
;nrailel  to  BA. 
The  /    DEF  =  L  ABC. 


A  TREATISE   ON   A   BOX    OF 


3.  To  find  a  third  proportional  to  two  given  straight 
lines,  AB,  AC. 


Place  them  together  so  as  to  form  any  angle  DAE. 
Take  AD  =  AC.     Draw  BC,  and  DE,  parallel  to  it. 
AB  :  AC  : :  AC  :  AE. 

4.  To  find  a  fourth  proportional  to  three  given  straight 
lines  AB.  CD,  EF. 


(i  K  Al 

Make  any  angle,  LGM.    Take  GH  =  AB,  GK  =  CD, 
and  GL  =  EF      Join  HK,  and  draw  LM  parallel  to  it 

AB  :  CD  : :  EF  :  GM. 
For  GH  :  GK :  :  GL  :  GM. 

5.  Another  method.     Given  AB  :  CD  :  :  EF  :  ? 
K 


INSTRUMENTS   AND   THE   SLIDE-RULK. 


29 


Make  a  rectangle  GITLK,  of  the  second  and  third, 
CD,  EF ;  and  in  one  of  the  sides,  produced  if  necessary, 
take  KM,  equal  to  the  first,  AB.  Draw  GM,  to  meet 
HL,  produced  if  necessary,  in  N. 

AB  :  CD  : :  EF  :  HN. 
For  KM  :  GH  : :  KG  :  HN. 

And  the  third  problem  may  be  performed  in  a  similar 
manner,  by  making  a  square  of  the  second  term 

6.  Hence  to  inscribe  a  square  in  a  given  triangle  ABC. 
AD  E 


B      K  L      C 

Through  the  vertex  A,  parallel  to  BC,  draw  the  straight 
line  AE,  and  from  C  raise  a  perpendicular  to  meet  it  in  D. 
Draw  DE,  equal  to  DC.  Join  EB,  cutting  DC  in  F. 
Through  F  draw  FG,  parallel  to  BC,  and  through  H  and 
G  draw  HL,  GK,  parallel  to  DC.  GL  will  be  the 
square  required. 

For  FC  :  CB  : :  CD  :  BC  +  DE  rB     rTj 

That  is,  GH  :  CB  : :  CD  :  BC  +  CD .-.  GH  ^ 


30 


A   TREATISE   ON   A   BOX   OF 


That  is,  the  side  of  the  inscribed  square  is  equal  to  the 
product  of  the  base  and  altitude  divided  by  their  sum  :  or 
the  sum  is  a  fourth  proportional  to  the  side,  base,  and 
altitude. 

7.  To  divide  a  given  line  AB,  similarly  to  another  CD 


A- 

On  CD,  construct  the  equilateral  triangle  CDE,  and 
.'rom  the  vertex  downwards  cut  off  EF  —  AB.  Draw  FG, 
parallel  to  CD,  and  join  EH,  EK.  Transfer  the  divisions 
L  and  M,  to  0  and  P.  AB  is  divided,  in  the  points  OP, 
similarly  to  CD,  in  H  and  K,  that  is, 

CD  :  AB  :  :  CK  :  AP  :  :  CH  :  AO. 

8.  Another  method.  Let  AB  be  the  divided  line,  and 
AC  the  line  required  to  be  similarly  divided. 

Lay  them  together,  making  any  angle  CAB.  Join 
the  extremities  CB,  and  draw  GE,  FD,  parallel  to  CB. 
AC  is  divided  similarly  to  AB ;  that  is, 

AB  :  AC  :  :  AE  :  AG  :  :  AD  :  AF. 

C 


D     E 


INSTRI'MKNT?!    AND    THE    SLIDE-RULE. 


31 


9.  Hence  to  divide  a  line  AB,   into  any  number  of 
equal  parts,  as  four. 


Draw  an  indefinite  line  CD,  and  from  C,  set  off  any 
Distance  the  intended  number  of  times,  in  the  points 
1,  2,  3, 4.  On  C  4,  construct  the  equilateral  triangle  EC  4. 
Make  EF  =  AB,  and  draw  FG,  parallel  to  CD.  FG  is 
equal  to  AB.  Join  E  1,  E  2,  E  3  j  and  FG,  that  is  AB, 
w'Jl  be  divided  into  four  equal  parts. 

10.  Or  by  the  other  method. 


E     D     B 


From  A,  draw  the  straight  line  AC,  making  any  xngle 
CAB.  From  A,  set  off  any  distance  the  intended  num- 
ber of  times  toward  C,  in  the  points  1,  2,  3,  4.  Draw  the 
line  4  B,  and  parallel  to  it  3  D,  2  E,  1  F.  AB  will  be 
divided  equally  into  the  required  number  of  parts. 


A   TREATISE   ON   A  BOX   OP 


11.  To  reduce  a  trapezium  ABCD,  to  a  triangle 


Draw  the  diagonal  AC,  and  through  B  draw  BE; 
*»arallel  to  it,  meeting  DC,  produced  to  E.  Join  AE. 
The  triangle  ADE,  iy  equal  to  the  trapezium  ABC. 

12.  To  reduce  any  polygon  ABCDE,  to  a  triangle. 
C 

13    "  //  \\~~~ D 


Draw  the  diagonals  CE,  CA,  and  produce  AE,  both 
ways,  to  F  and  Gr.  Draw  DG-,  parallel  to  CE,  and  BF, 
to  CA.  Join  CF,  CG.  The  triangle  CFG,  is  equal  to 
the  polygon  ABCDE. 

13.  To  reduce  a  triangle  ABC,  to  a  parallelogram. 

E 
F 


INSTRUMENTS    AXD    THE    SLIDE-RULE. 


33 


Through  the  vertex  A  draw  FA,  parallel  to  BC.  Bisect 
BC  in  D,  and  raise  DE,  perpendicular  to  BC,  meeting 
FA  in  E.  Draw  FB  parallel  to  ED.  The  redangle 
FD.  is  equal  to  the  triangle  ABC,  for  the  content  of  a 
triangle  is  equal  to  the  product  of  half  the  base  by  the 
altitude. 

14.  To  reduce  a  parallelogram  ABCD,  to  a  square. 
A  D 


Bi- 


Produce BC  to  E,  making  CE  =  DC.  Bisect  BE 
in  F.  Produce  DC  to  K,  making  FK  =  FE,  and 
CG  =  CK.  Through  K  draw  KH,  parallel  to  BE,  and 
through  G  draw  GH,  parallel  to  DK.  The  square  GK, 
is  equal  to  the  rectangle  AC,  and  CG  is  a  mean  propor- 
tional between  DC  and  CB  j  that  is, 

DC  :  CG  :  :  CG  :  CB. 


15.  Hence  to  make  a  square  equal  to  any  given  polygon 
ALDC. 


34 


A   TREATISE   ON   A   BOX   OF 

B 


Reduce  it  to  the  triangle  ACE,  and  this  to  the  paral- 
lelogram CHGF,  and  this  to  the  square  NM. 

16.  To  reduce  a  triangle  ABC,  to  another  that  shall 
be  of  a  given  altitude  AD. 


Join  DC,  and  through  the  vertex  B  draw  BE,  parallel 
to  the  base  AC,  meeting  AD,  produced  if  necessary,  in 
the  point  E.  Through  E  draw  EF,  parallel  to  DC. 
Join  DF.  The  triangle  DAF  =  ABC. 


INSTRUMENTS   AND   THE   SLIDE-RULE. 


35 


THE  PROTRACTOR. 

It  is  unnecessary  to  describe  the  construction  of  this. — 
It  is  simply  a  semicircle,  divided  into  180  equal  parts, 
termed  degrees.  As  mentioned  in  the  introduction,  these 
degrees  are,  in  some  boxes,  transferred  to  the  border  of  the 
plain  scale,  which  is  used  precisely  as  the  protractor :  it  is, 
however,  far  from  being  so  convenient  as  the  semicircle. 
Some  protractors  are  complete  circles,  and  contain,  of  course 
360  degrees. 

USES. 

1.  To  find  the  number  of  degrees  contained  in  any 
given  angle  BAG. 


Lay  the  central  notch  of  the  instrument  upon  A,  and 
the  edge  along  AB,  as  in  the  diagram ;  and  observe  the 
number  cut  by  the  other  line  AC. 

2.  To  lay  down  an  angle  ABC,  which  shall  contain  a 
given  number  of  degrees. 


36 


A  TREATISE   ON   A   BOX   OF 


B  C 

Draw  a  line  BC,  of  any  length.  Place  the  notch  against 
B,  and  the  edge  along  BC.  Prick  a  point  D  against  the 
number  required,  and  through  it  draw  the  line  AB. 

3.  Through  a  given  point  C,  to  draw  a  line  paralell  to 
a  given  line  AB. 

D  C 


In  AB  take  any  point  E,  and  join  CE;  and  make  the 
angle  DCE,  equal  to  the  angle  CEB,  by  the  line  DC. 
DC  is  parallel  to  AB. 

4.  To  divide  a  given  line  AB,  into  any  number  of 
equal  parts. 


INSTRUMENTS   AND   THE   SLIDE-RULE. 


37 


From  the  extremities  of  the  line  AB,  draw  the  lines 
AC,  BD,  making  equal  angles.  From  A  towards  C,  and 
from  B  towards  D,  set  off  any  distance  once  less  than  the 
intended  number  of  parts.  Number  one  from  the  extre- 
mity A,  and  the  other  towards  the  other  extremity  B,  and 
join  the  like  numbers.  AB  will  be  divided  as  required. 

5.  To  erect  a  perpendicular  to  a  given  line  AB,  from 
a  point  C,  within  it. 

D 


Place  the  edge  along  A  B,  and  the  notch  at  C.  Then 
against  90  prick  the  point  D,  and  draw  DC.  DC  will  be 
the  perpendicular  required. 

6.  To  let  fall  a  perpendicular  upon  a  straight  line  AB 
from  a  point  C. 


Draw  any  line  CA.  Observe  the  number  of  degrees 
contained  in  the  angle  CAB.  Subtract  it  from  90,  and 
make  the  angle  ACB,  equal  to  the  remainder,  by  the  line 
CB.  CI>  will  be  the  perpendicular  required. 


38 


A   TREATISE   ON    A   BOX    OF 


7.  To  divide  a  given  angle,  ABC,  into  any  number  of 
equal  parts. 


Find  the  number  of  degrees ;  divide  it  by  the  required 
number  of  parts,  and  prick  off  the  quotient  along  the  rim, 
as  in  D.  Join  BD. 

8.  To  inscribe  a  circle  in  a  given  triangle,  ABC. 


Bisect  any  two  of  the  angles  ABC,  ACB,  by  the 
straight  lines  BD,  CD,  crossing  each  other  in  D,  from 
which  let  fall  the  perpendicular  DE.  DE  is  the  radius 
of  the  required  circle. 


INSTRUMENTS   AND   THE    SLIDE-RULE. 

9.  In  a  circle  to  inscribe  any  regular  polygon. 


39 


Divide  360  by  the  intended  number  of  sides.  Place  the 
instrument  with  the  notch  against  the  centre,  and  prick  off 
the  quotient  round  the  rim.  If  the  number  of  sides  be 
odd,  the  instrument  will  require  to  be  turned  round ;  if 
even,  half  may  be  pricked  off,  and  lines  drawn  through 
the  centre,  the  extremities  of  which  meeting  the  circle, 
will  give  the  points  required.  Connect  the  points,  and  the 
polygon  will  be  completed. 

]0.  To  construct  a  regular  polygon  on  a  given  line.AIJ. 


40 


A    llvKATlSE   CH    A  BOX   OP 


Divide  360  by  the  intended  number  of  sides,  subtract 
the  quotient  from  180,  and  halve  the  remainder.  Make 
the  angles  CAB,  CBA,  each  equal  to  this,  by  lines  inter- 
secting in  C.  "  CA  is  the  radius  of  a  circle,  round  which 
the  line  AB  may  be  carried  the  required  number  of  times. 

11.  To  describe  a  circle  within  or  without  a  regular 
polygon. 


Bisect  any  two  angles,  ABC,  BCF,  by  the  lines  BD, 
CD,  and  from  D  let  fall  DE,  perpendicular  to  the  side 
BC.  BD  is  the  radius  of  the  outer,  and  ED  of  the  inner 
circle,  as  required.* 


THE  PLAIN  SCALE. 

THE  method  of  constructing  the  plain  scale  is  obvious. 
A  number  of  horizontal  lines  on  one  side  having  been 
drawn  through  the  whole  length  of  the  rule,  and  a  vertical 
column  on  the  left  for  the  reception  of  numbers,  a  distance 
of  two  inches  is  laid  down,  and  divided,  upon  the  top  line 

*  Polygons  are  more  expeditiously  constructed  by  means  of  the  Sector,  of 
which  hereafter. 


INSTRUMENTS   AND   THE   SLIDE-RULE. 


41 


into  9  equal  parts,  upon  tlie  next  into  8,  and  so  on,  down 
to  3.  These  are  repeated  along  the  rule  as  often  as  ita 
length  will  allow ;  the  first  portion  of  each  is  subdivided 
into  tenths  and  twelfths  ;  and  numbers  are  placed  in  the 
column,  showing  how  many  of  the  tenths  are  contained  in 
an  inch.  A  scale  of  chords,  marked  C,  is  graduated  along 
the  top ;  and  at  60  a  small  r  is  placed,  indicating  that 
distance  from  the  beginning  to  be  the  radius  of  the  circle 
from  which  they  are  taken.  They  are  merely  the  degrees 
of  a  quadrant  or  quarter  of  a  circle,  laid  down  in  a  straight 
line ;  thus, — 


C  B 

Draw  the  lines  B  A,  BC,  at  right  angles  to  each  other ; 
and  with  any  convenient  distance,  BA,  describe  the  arc 
AsC ;  divide  it  into  90  equal  parts,  and  join  AC.  From 
C  as  a  centre,  with  the  distances  C  10,  C  20,  &c.,  describe 
the  arcs  10,  107;  20,  20',  &c.  meeting  the  line  AC.  Fill 
up  the  separate  degrees,  which  are  not  marked  in  the 
diagram  to  prevent  confusion,  and  the  scale  is  complete- 


42  A  TREATISE  ON   A  BOX   OP 

It  is  evident,  by  inspection,  that  the  chord  of  60  is  equal 
to  the  radius,  as  shown  by  the  letter  r  upon  the  rule ; 
which  distance  is  therefore  always  to  be  taken  in  laying 
down  angles,  as  will  be  described  when  we  come  to  speak 
of  its  uses. 

The  other  face  of  the  rule  is  divided  along  the  top  into 
inches,  and  these  into  tenths.  The  next  line  is  divided 
into  50  equal  parts.  Under  these  is  what  is  called  the 
diagonal  scale.  It  consists  of  11  equidistant  parallel 
lines,  crossed  by  vertical  ones  a  quarter  of  an  inch  apart. 
By  taking  these  alternately,  another  scale  is  obtained,  of 
twice  the  size  of  the  former.  The  first  of  each  of  these  is 
divided  into  ten  equal  parts,  above  and  below ;  and  oblique 
lines  are  drawn  from  the  first  perpendicular  below  to  the 
first  division  above,  and  continued  parallel,  by  which  con- 
trivance the  first  space  is  divided  into  100  equal  parts: 
consequently,  if  the  line  contain  ten  of  the  large  divisions, 
each  of  these  small  spaces  is  the  thousandth  part  of  such 
line.  If,  therefore,  the  large  divisions  denote  hundreds, 
the  first  subdivisions  will  be  tens,  and  the  second,  units. 


USES. 

The  plain  scale  is  simply  for  laying  off  distances.  The 
manner  of  using  the  first  side  is  evident.  If  the  number 
47  be  required,  place  one  foot  of  the  compasses  upon  4, 
and  extend  the  other  to  the  7th  subdivision  of  the  tenths. 
If  3  feet  5  inches  be  required,  place  one  foot  on  3,  and 
stretch  the  other  to  the  5th  division  of  the  twelfths.  The 
following  figure  is  laid  down  from  the  scale  at  the  bottom, 
numbered  15,  and  may  be  tested  by  the  pupil. 


INSTRUMENTS   AND   THE   SLIDE-RULE. 
.  !<),€». 


43 


The  scale  of  chords  serves  the  purposes  of  the  protractor, 
and  is  used  as  follows : — 

1.  To  find  the  number  of  degrees  contained  in  a  given 
angle,  BAG. 


From  A  with  the  radius  60  describe  the  arc  a  a.  Take 
the  distance  from  a  to  a  in  the  compasses,  and  apply  it  to 
the  beginning  of  the  scale.  The  number  to  which  it 
reaches,  shows  the  degrees  contained  in  the  angle. 

2.  To  make  an  angle  ABC,  which  shall  contain  a  given 
number  of  degrees,  as  26. 


From  B  with  the  radius  GO  describe  the  arc  AC.     Take 
26  from  the  scale ;  place  one  foot  of  the  compasses  in  C 


44  A  TREATISE   ON  A   BOX   OP 

and  with  the  other  cross  the  arc  in  A      Join  AB.     ABC 
is  the  angle  required. 

And  in  the  same  manner  may  be  performed  all  the  pro- 
blems described  under  the  protractor,  which  need  not  be 
repeated  here. 

The  diagonal  scale  is  used  where  more  exactness  is  re- 
quired, or  when  a  number  containing  three  figures  is 
wanted,  as  357,  35.7,  3.57,  &c.  If  the  primary  divisions 
denote  hundreds,  the  subdivisions  express  tens,  while  the 
units  are  counted  on  the  parallels — upwards  on  the  left, 
the  quarter-inch  scale ;  and  downwards  on  the  right,  the 
half-inch  scale.  If  the  primary  divisions  (those  denoted 
by  the  perpendiculars)  express  tens,  the  diagonals  will  be 
units,  and  the  parallels  tenths,  and  so  on ;  each  smaller 
division  being  the  tenth  of  the  next  larger. 

In  taking  off  numbers  from  this  scale,  proceed  in  an 
inverse  order  to  the  figures;  that  is,  commence  with  the 
units,  proceed  to  the  tens,  and  end  with  the  hundreds; 
thus,  to  take  off  346  from  the  larger  scale.  Place  one 
foot  of  the  compasses  upon  the  sixth  parallel  where  it  is 
crossed  by  the  fourth  diagonal,  and  extend  the  other  to 
the  third  perpendicular.  To  take  off  1839  from  the 
smaller  scale.  On  the  9th  parallel,  where  it  is  crossed 
by  the  third  diagonal,  place  one  foot  of  the  compasses, 
and  extend  the  other  to  the  18th  perpendicular. 
To  raise  a  perpendicular  to  a  given  line,  AB. 

Make  AC  =  4  equal  parts. 
From  C,  with  a  distance  of  3 
from  the  same  scale,  make 
an  arc ;  and  from  A  with  5 
cross  it  in  D.  Join  DC- 
-D  DC  will  be  the  perpendicu- 
lar required. 


INSTRUMENTS   AND   THE    SLIDE-RULE. 


Given  two  square  pieces  of  deal  board ;  the  side  of  one 
17  inches,  of  the  other  29.  It  is  required  to  ascertain  the 
side  of  another  that  shall  be  equal  to  both. 


Lay  down  a  base,  AC  =  29,  and  raise  a  perpendicular 
BC  =  17.  Join  AB;  apply  it  to  the  scale,  and  it  will  be 
found  33.6.  For  the  square  of  the  hypothenuse  is  equal 
to  the  sum  of  the  squares  of  the  base  and  perpendicular. 

It  is  required  to  find  the  diameter  of  a  copper,  that, 
being  of  the  same  depth  as  another  whose  width  is  13 
inches,  may  contain  thrice  as  much. 


Make  AB  =  13,  and  raise  a 
perpendicular  AC.  From  B, 
with  twice  the  distance,  cross 
it  in  C.  Apply  CA  to  the 
scale;  it  will  be  found  to  be 
22  *.  For  if  AB  =  1,  and 
BC  =  2,  then  AC  = 


Three  men  bought  a  grindstone,  20  inches  in  diameter ; 
and  agreed  that  the  first  should  use  it  till  he  ground  down 
J  of  it  for  his  share  ;  the  second  to  do  the  same ;  and  the 
third  to  finish  the  remainder.  Ascertain  the  diameters  of 
the  second  and  third  shares. 


4G 


A   TREATISE    ON    A   BOX    OP 


Draw  the  line  AB  =  20 ;  bisect  it  in  C,  and  on  AC 
describe  the  semicircle  AEDC.  Divide  AC  into  three 
equal  parts,  in  the  points  FG- ;  perpendicular  to  which 
draw  the  lines  EF,  DGr,  to  meet  the  semicircle.  Join 
EC,  DC,  and  produce  them  till  CH  be  equal  to  CD,  and 
CK  to  CE.  EK  is  the  diameter  for  the  second  person, 
and  DH  for  the  third.  By  applying  them  to  the  scale, 
EK  will  be  found  to  be  about  16J,  and  DH  rather  more 
than  11}.  For  AC.CG  ==  CD3  and  AC.CF  =  CES 
.-.  GG  :  CF  : :  CDa  :  CE3;  and  the  areas  of  circles  are 
as  the  squares  of  their  radii. 

Four  men  bought  a  grindstone  of  30  inches  in  diameter; 
and  agreed  that  the  first  should  use  it  till  he  ground  down 
l-4th  of  it  for  his  share,  deducting  6  inches  in  the  middle 
for  waste ;  and  then  that  the  second  should  use  it  till  he 
ground  down  l-4th  part ;  and  so  on.  What  part  of  the 
diameter  must  each  grind  down  ? 

l-5th  of  the  diameter  being  waste,  125th  of  the  content 


INSTRUMENTS    AND    THE    SLIDE-RULE. 


is  waste ;  therefore,  conceiving  the  whole  to  contain  25 
shares,  1  share  will  be  for  waste,  and  each  person  will  have 
6  shares. 

Hence,  draw  the  line  AB  =  30,  and  on  AC  describe  the 
semicircle  AHKLMC.  Divide  AC  into  25  equal  parts 
by  problem  10,  parallel  ruler;  and  make  AG-,  GF,  FE, 
ED,  each  equal  to  6  of  these  parts.  From  the  points 
G,  F,  E,  D,  raise  perpendiculars  to  meet  the  semicircle  in 
H,  K,  L,  Mj  join  HC,  KG,  LC,  and  MC;  and,  having 
drawn  circles  from  H,  K,  L,  M,  with  C  as  a  centre,  produce 
them  to  N,  0,  P,  Q.  The  diameter  AB  is  30  :  HQ  will 
be  found,  upon  applying  it  to  the  scale,  to  measure  about 
26;  KP,  nearly  21 J;  LO,  about  16;  and  MN,  6  for  the 
waste.  Subtracting  these  in  succession,  we  have  4  inches 
for  the  first,  4$  for  the  second,  5£  for  the  third,  and  10 
for  the  last. 

The  student  will  find  these  problems  repeated  at  the 
end  of  the  Uses  of  the  Slide  Rule. 


48 


A  TREATISE   ON  A   BOX  OP 


TRIGONOMETRY. 

THE  circumference  of  a  circle  is  supposed  to  be  divided 
into  360  equal  parts,  called  degrees.  Each  degree  is 
divided  into  60  minutes;  each  minute  into  60  seconds; 
and  so  on. 

Degrees  are  marked  with  a  small  °  at  the  top  of  the 
figure,  minutes  with  ',  seconds  with  ",  and  so  on.  Thus, 
36°  18'  25"— 36  degrees,  18  minutes,  25  seconds. 


The  difference  of  an  arc  from  90  degrees,  or  a  quarter, 
is  called  its  complement ;  thus  FC  is  the  complement  of 
CB.  The  chord  of  an  arc  is  a  line  drawn  from  one  ex- 
tremity of  the  arc  to  the  other ;  thus  CK  is  the  chord  of 
the  arc  CBK. 

The  sine  of  an  arc  is  a  line  drawn  from  one  extremity 
of  the  arc  perpendicular  to  the  diameter  passing  through 
the  other  extremity ;  thus  CD  is  the  sine  of  the  arc  CB, 
or  angle  CAB,  which  it  measures ;  and  CE  is  the  sine  of 
the  arc  CF,  or  angle  CAF,  which  it  measures. 

The  sine  is  half  the  chord  of  twice  the  arc,  or  angle. 


INSTRUMENTS   AND   THE   SLIDE-RULE.  49 

The  tangent  of  an  arc  is  a  line  touching  the  circle  in 
»ne  extremity  of  the  arc,  and  meeting  a  line  drawn  from 
the  centre  through  the  other  extremity ;  thus  GB  is  the 
tangent  of  the  arc  CB,  or  angle  CAB;  and  FH  is  the 
tangent  of  the  arc  CF,  or  angle  CAF. 

The  secant  of  an  arc  is  the  line  meeting  the  tangent ; 
thus  GA  is  the  secant  of  the  arc  CB,  or  angle  CAB  j  and 
AH  is  the  secant  of  the  arc  CF,  or  angle  CAF. 

The  versed  sine  of  an  arc  is  the  part  of  the  diameter 
intercepted  between  the  arc  and  its  sine  :  thus  DB  is  the 
versed  sine  of  CB. 

The  cosine,  cotangent,  and  cosecant  of  an  arc,  are  the 
complement's  sine,  tangent,  and  secant :  co  being  simply 
a  contraction  of  complement.  Thus  CE,  or  AD,  is  the 
cosine  of  CB,  being  the  sine  of  the  complement  CF :  so 
FH  is  the  cotangent  of  the  arc  CB,  being  the  tangent  of 
the  complement  FC;  and  AH  is  the  cosecant  of  CB, 
being  the  secant  of  the  complement  CF. 

From  these  definitions  it  is  evident — 

1st.  That  when  the  arc  is  0,  the  sine  and  tangent  are  0, 
but  the  secant  is  then  the  radius  AB. 

2d.  When  the  arc  is  a  quadrant,  FB,  then  the  sine  is  the 
greatest  it  can  be,  being  the  radius  of  the  circle ;  aad  the 
tangent  and  secant  are  infinite. 

3d.  The  versed  sine  and  cosine  together  make  up  the 
radius. 

4th.  The  radius  AB,  the  tangent  BG,  and  secant  AG, 
form  a  right-angled  triangle. 

5th.  The  cosine  AD,  the  sine  DC,  and  radius  AC,  also 
form  a  right-angled  triangle. 

6th.  The  radius  AF,  the  cotangent  FH,  and  cosecant 
AH,  also  form  a  right-angled  triangle.  And  since  the 

5 


50 


A   TREATISE   ON   A   BOX   OF 


angle  FAH  =  the  angle  ACD  =  the  angle  AGB; 
these  right-angled  triangles  are  similar  to  each  other 
Hence 


AD 

DC 

AB 

BG 

viz.  cosine 

sine 

radius 

tangent. 

.  AE 

EG 

AF 

FH 

viz.     sine 

cosine 

radius 

cotangent. 

AD 

AC 

AB 

AG. 

viz.  cosine 

radius 

radius 

secant. 

EA 

AC 

FA 

AH. 

viz.     sine 

radius 

radius 

cosecant. 

GB 

BA 

AF 

FH. 

viz.  tangent 

radius 

radius 

cotangent. 

So  the  radius  is  a  mean  proportional  between  the  cosine 
and  secant,  the  sine  and  cosecant,  and  the  tangent  and 
cotangent. 

In  every  case  in  trigonometry  three  parts  must  be  given 
to  find  the  other  three ;  and  one  of  these,  at  least,  must  be 
a  side. 

The  cases  in  trigonometry  are  of  three  varieties : 

1st.  When  a  side  and  its  opposite  angle  are  given. 

2d.  When  the  two  sides  and  the  contained  angle  are 
given. 

3d.  When  the  three  sides  are  given. 


When  a  side  and  its  opposite  angle  are  two  of  the  given 
parts  j  then 

Any  side  :  sine  of  its  opp.  angle  :  :  any  other  side  :  sine 
of  its  opp.  angle. 


INSTRUMENTS    AND   THE    SLIDE-RULE. 


51 


And, 


Sine  of  any  Z_  :  its  opp.  side  :  :  sine  any  othei  L.  : 
its  opp.  side. 


Make  AE  =  BC,  and  EF,  BD,  perpendicular  to  AC. 
Then  AB  :  BD  :  :  AE  :  EF.     But  AE  =  BC; 

.-.  AB  :  BD  :  :  BC  :  EF. 
But  BD  is  the  sine  of  C,  and  EF  of  A ; 

.-.  AB  :  sin  opp.  Z_  C  :  :  BC  :  sin.  its  opp.  l_  A. 


n. 

When  two  sides  and  their  contained  angle  are  given. 
Sum  of  sides  :  their  difference  :  :  tang.  \  sum  of  oppo- 
site angles  :  tang,  of  J  their  diff. 
Then  \  sum  -{-  i  diff.  =  greater ; 

and  \  sum  —  \  diff.  =  less. 

Let  ABC  be  the  proposed  triangle,  having  the  two 
given  sides  AC,  BC,  including  the  given  angle  C. 


From  C,  with  radius  CA,  describe  a  semicircle,  meeting 


62  A   TREATISE   ON    A   BOX   OP 

BC  produced  in  D  and  E.  Join  AE,  AD,  and  draw  DF 
parallel  to  AE. 

Then  BE  =  BC  +  CA,  the  sum  of  the  sides  BC,  CA: 
and  BD  =  BC  —  CA,  their  difference. 

Also,  CAB  +  CBA  =  CAD  +  CDA  =  2CDA ; 

.-.  L  CDA  =1 1  CAB  +  CBA.  j 

That  is,  CDA  is  half  the  sum  of  the  unknown  angles. 
Again,  L  CDA  =  L  DBA  +  L  DAB ; 
.-.  f_  DAB  =  L  CDA  —  f_  DBA 
=  L  CAD  —  L  CBA, 
Add  to  each  side  DAB  ;  then, 

2  DAB  =  CAD  -f  DAB  —CBA 
=  CAB  —  CBA ; 

.-.  DAB  =  i  |  CAB  —  CBA.  j 

That  is,  DAB  is  half  the  difference  of  the  unknown 
angles. 

Now  EAD  being  a  semicircle,  EA  is  perpendicular  to 
AD,  as  is  also  DF;  .•.  AE  is  the  tangent  of  CDA,  and 
DF  the  tangent  of  DAB,  to  the  same  radius  AD. 

But,  BE  :  BD  :  :  AE  :  DF; 
that  is,  the  sum  of  the  sides  :  difference  of  the  sides  :  : 

tangent  of  \  sum  of  opposite  angles  :  tangent  of  \  their 

difference. 

Three  angles  of  a  triangle  being  equal  to  180°,  the  sum 
of  the  unknown  angles  is  found  by  taking  the  given  angle 
from  180°. 

in. 
When  the  three  sides  are  given,  to  find  the  angles. 


INSTRUMENTS    AND   THE   SLIDE-RULE, 


53 


Base  :  sum  of  other  sides  :  :  difference  of  those  sides 
:  difference  of  segments  of  base,  made  by  perpendicular 
falling  from  the  vertex. 
Let  ABD  be  the  given  triangle. 


From  B  with  the  distance  BA  describe  a  circle,  and 
produce  DB  to  G-.  Then, 

DG  =  DB  +  BA;  &nd  HD  =  DB  —  BA, 

Also,  since  AE  =  EF ;  .  •.  FD  =  DE  —  EA. 

But  AD  :  DG  :  :  HD  :  FD,  (Euclid  3,  46 ;)  that  is, 
Base  :  sum  of  other  sides  :  :  diff.  of  those  sides  :  diff.  of 
segments  of  base. 

Hence,  in  each  of  the  two  right-angled  triagles,  there 
will  be  known  two  sides,  and  the  right  angle  opposite  to 
one  of  them. 

All  cases  of  plane  triangles  may  be  solved  by  these 
three  problems  ;  but  for  right-angled  triangles,  the  follow- 
ing are  more  convenient : — 


54 


A   TREATISE   ON   A   BOX    OF 


First  make  the  base  AB  radius ;  then, 

Radius  :  tang.   A  :  :  AB  :  BC ;  and 
Radius  :  secant  A  :  :  AB  :  AC. 

Next  make  CB  radius. 


Then  Radius  :  tang.    C  : 
Radius  :  secant  C   : 
Lastly,  make  AC  radius. 


:  CB  :  BA;  and 
:  CB  :  CA. 


\ 


Then, 


Radius  :  sine  A  :  :  AC  :  CB ;  and 
Radius  :  cos.  A  :  :  AC  :  AB. 


SECTOR. 

THE  lines  on  the  sector,  as  before  stated,  are  of  two 
sorts,  single  and  double.  Only  the  double  lines  are  sec- 
toral ;  these  proceed  from  the  centre,  and  are — 

1st.  A  scale  of  equal  parts,  marked  L,  and  containing 
'.00  divisions. 

°d.  A  scale  of  chords,  marked  C,  running  to  60. 


INSTRUMENTS   AND   THE   SLIDE-RULF..  55 

3d.  A  line  of  secants,  marked  S,  running  to  75. 

4th.  A  line  of  polygons,  marked  POL.  These  are  num- 
bered backwards  from  4  to  12. 

Upon  the  other  face,  the  sectoral  lines  are, — 

1-:.  A  line  of  sines,  marked  S,  running  to  90.  The 
sines  may  be  easily  distinguished  from  the  secants,  though 
marked  the  same ;  as  the  distances  of  the  sines  diminish 
towards  the  end,  while  the  secants  increase. 

2d.  A  line  of  tangents,  marked  T,  running  to  45. 

3d.  Between  the  line  of  tangents  and  sines  there  is 
another  line  of  tangents,  beginning  at  a  quarter  of  the 
length  of  the  former,  to  supply  their  defect,  and  extend- 
ing from  45  to  75,  marked  t  or  T. 

The  distance  from  T  to  T,  from  S  to  S,  from  C  to  C, 
and  from  L  to  L,  is  the  same ;  so  that  at  whatever  distance 
the  sector  may  be  opened,  the  angles  formed  by  those  lines 
will  always  be  equal.  The  polygons  are  laid  down  to  a 
shorter  radius,  for  the  sake  of  including  the  pentagon  and 
square.  The  radius,  the  chord  of  60°,  the  sine  of  90°, 
the  tangent  of  45°,  the  secant  of  0°,  all  are  equal. 

The  method  of  constructing  the  sectoral  lines  is  exhi- 
bited in  Fig.  1,  fronting  the  title-page. 

From  the  point  A  with  any  convenient  radius  describe 
the  circle  BODE,  and  draw  the  diameters  BD,  CE,  cross- 
ing each  other  at  right  angles  in  the  centre.  Produce  C 
to  F,  and  through  B  draw  BG,  a  tangent  to  AB.  Join 
EB,  BC,  CD. 
In  the  construction  of  the  following  Scales,  only  the  primary 

dii-isions  are  drawn,  the  smaller  ones  being  omitted  to 

prevent  confusion  : — 

Divide  AD  into  ten  equal  parts,  and  these  again  into 
tenths ;  so  shall  AD  be  a  line  of  equal  parts. 


66  A   TREATISE   ON    A   BOX   OP 

Divide  the  arc  BC  into  9  equal  parts,  and  these  again 
into  tenths,  and  with  the  compasses  from  B,  as  a  centre, 
transfer  the  divisions  to  the  straight  line  BC;  so  shall  BG 
be  a  line  of  chords. 

From  each  of  the  divisions  of  the  arc  BC  let  fall  perpen- 
diculars upon  AB,  and  number  them  backwards ;  so  shall 
AB  be  a  line  of  sines. 

From  the  point  A,  through  the  divisions  of  the  arc  BC, 
draw  straight  lines  to  meet  BG;  so  shall  BG  be  a  line  of 
tangents.  And  from  D  to  the  same  divisions  of  the  arc 
BC,  draw  straight  lines  cutting  AC ;  so  shall  AC  be  a  line 
of  half-tangents. 

The  distances  from  the  centre  A  to  the  divisions  on  the 
line  of  tangents  being  transferred  to  AF,  AF  will  be  a  line 
of  secants. 

From  B,  in  the  arc  BE,  cut  off  the  fifth  part  of  the 
circumference,  72°,  and  transfer  it  with  the  compasses  to 
the  straight  line  BE.  Do  the  same  with  the  sixth, 
seventh,  eighth,  &c.  ;  so  shall  BE  be  a  line  of  polygons. 
Transfer  these  distances  to  the  lines  upon  the  rule,  and 
the  sectoral  part  is  complete.* 

From  the  property  of  similar  triangles,  AB  :  BC  :  : 
Ab  :  be;  hence,  if  AB  were  divided  into  ten  equal  parts, 
and  A.b  contained  four  of  those  parts,  then  BC  being  di- 
vided into  ten  equal  parts,  be  would  contain  four  of  those 
parts. 

And  if  AB  were  the  sine  of  90°,  and  Ab  the  sine  of 


*  These  lines  are  best  laid  down  by  the  help  of  tables  of  natural  sines, 
tangents,  <tc.,  in  the  same  way  as  the  lines  of  the  slide-rule  are  laid 
down  by  logarithmic  numbers,  sines,  and  tangents ;  of  which  here- 


INSTRUMENTS    AND   THE   SLIDE-RULE. 

B 


57 


40° ;  then  BC  being  taken  the  sine  of  90°,  be  would  be 
the  sine  of  40°. 

And  if  AB  were  the  radius  of  a  circle,  and  A6  the  side 
of  an  octagon  inscribable  iu  the  same  ;  then  BC  being  the 
radius  of  another  circle,  be  would  be  the  side  of  aa  octagon 
inscribable  in  the  same. 

And  hence,  though  the  lateral  scale  AB  or  fs*'A,  yet  a 
parallel  scale  BC  is  obtainable  at  pleasure. 

The  manner  of  taking  distances  from  th:  vy///rrJ  lines 
will  be  best  understood  from  the  frliowir//,  />gJ-r}}  «ri*'ch 
contains  a  portion  of  the  line  L. 


When  the  distance  is  taken  from  the  centre  1  ,  along 


58  A   TREATISE   ON   A   BOX   OP 

either  of  the  legs  to  any  point,  BC.  or  D,  it  is  called  a 
lateral  distance ;  when  it  is  taken  from  any  point  of  one 
leg  to  the  corresponding  point  of  the  other  leg,  as  from  B 
to  B,  from  C  to  C,  from  D  to  D,  or  from  L  to  L,  it  is 
Called  a  parallel  distance.  To  make  any  length  then,  aa 
for  instance  an  inch,  a  parallel  distance  from  3  to  3,  or  D 
to  D,  the  points  of  the  compasses  are  to  be  opened  aa 
inch ;  then  placing  one  point  against  the  3  of  one  leg  of 
the  rule,  open  the  rule  till  the  other  point  will  fall  upon 
the  3  of  the  other  leg.  Observe,  the  points  of  the  com- 
passes must  always  be  placed  upon  the  lines  nearest  the 
opening  of  the  rule — upon  those  which  would,  if  continued, 
meet  in  the  centre.  As  distances  are  taken  from  all 
the  sectoral  lines  in  the  same  manner,  and  as  they  are  of 
most  extensive  utility  when  used  conjointly,  it  will  not  be 
necessary  to  treat  of  them  separately. 


USES   OF   THE    SECTOR. 

To  divide  a  given  line  AB  into  any  number  of  equal 
parts,  as  6. 

A-        ...      . , , _«. ,B 

12  34          56 

Take  the  distance  from  A  to  B  in  the  compasses,  and 
make  it  a  parallel  distance  from  6  to  6  on  the  line  L; 
then  the  parallel  distance  from  1  to  1  will  be  the  sixth 
part  of  the  line  AB,  as  required. 

To  find  a  third  proportional  to  two  numbers,  4  and  6. 
Take  6  laterally,  and  make  it  a  parallel  distance  from  4 
to  4  on  L :  then  take  the  parallel  distance  from  6  to  6, 
and  apply  it  laterally  :  it  will  be  found  to  measure  to  9, 
the  third  proportional  required.  Or  ir  ake  the  lateral  dis- 


INSTRUMENTS   AND   THE    SLIDE-RULE.  59 

taiice  4,  a  parallel  distance  between  6  and  6 ;  then  will 
the  lateral  distance  6  be  found  a  parallel  distance  between 
9  and  9. 

To  find  a  fourth  proportional  to  three  given  numbers, 
3,  4,  and  (7.  Take  the  lateral  distance  3,  and  stretch  it 
as  a  parallel  from  4  to  4  on  L;  then  take  the  lateral  dis- 
tance 6,  it  will  be  found  to  extend  as  a  parallel  from  8  to 
8.  Or  make  4  a  parallel  distance  from  3  to  3j  then  will 
the  parallel  from  6  to  6  measure  laterally  to  8. 

The  reason  is  obvious ;  for  in  all  cases, — 
Any  lateral  distance  :  its  parallel  distance  :  :  any  other 

lateral  distance  :  its  parallel  distance. 
And  conversely, 

Any  parallel  distance  :  its  lateral  distance  :  :  any  other 
parallel  distance  :  its  lateral  distance. 

To  measure  the  lines  of  the  peri- 
c  meter  of  any  figure,  one  of  which, 
as  AB,  contains  a  given  number  of 
equal  parts,  as  4.  Make  AB  a  pa- 
rallel distance  from  4  to  4  on  L. 
Take  CB,  and  it  will  be  found  a 
parallel  distance  from  3  to  3 ;  take 
AC,  and  it  will  be  found  a  parallel 
distance  from  5  to  5. 

To  multiply  any  number,  3,  by  another,  7.  Make  the 
lateral  distance  3  a  parallel  distance  from  1  to  1  on  L; 
then  take  the  parallel  distance  from  7  to  7 ;  it  will  be 
found  the  lateral  distance  of  21. 

To  divide  any  number,  40,  by  another,  5.  Make  the 
parallel  distance  of  40  the  lateral  distance  of  5  on  L' 


60  A   TREATISE   ON   A    BOX   OF 

then  will  the  parallel  distance  of  1  be  the  lateral  distance 
of  8. 

To  divide  a  line  in  any  required  proportion,  as  2,  3,  and 
4.  Take  the  length  and  make  it  a  parallel  distance  front 
9  to  9,  their  sum,  on  the  line  L ;  then  will  the  parallel 
distance  from  2  to  2,  3  to  3,  and  4  to  4,  be  the  parts  re- 
quired. 

If  the  number  is  greater  than  100,  take  some  aliquot 
part  of  it,  and  then  multiply  the  result  by  the  number  by 
which  it  was  divided. 

To  measure  an  angle  ABC  with  the  line  of  chords,  C. 
With  any  radius  BA,  describe  the  arc  AC ;  make  BA  a 
parallel  distance  from  60  to  60  on  C ;  then  take  AC,  and 
moving  it  along,  find  the  numbers  to  which  it  will  apply 
as  a  parallel  distance. 


At  a  given  point  B,  in  a  given  line  AB,  to  make  an 
angle  containing  any  number  of  degrees,  as  30°. 


Open  the  sector  to  any  distance,  and  take  the  parallel 
distance  from  60  to  60,  with  which  describe  the  arc  DE. 
Take  the  parallel  distance  from  30  to  30 ;  and  setting  one 


INSTRUMENTS   AND   THE    SLIDE-RULE. 


61 


foot  of  tht  compasses  on  D,  prick  off  the  distance  to  E ; 
through  E  draw  BC.     ABC  is  the  angle  required. 

And  so  of  all  the  problems  given  under  the  protractor. 

To  measure  any  angle,  ABC,  with  the  line  of  sines. 

From  any  point,  A,  let  fall  the  line  AD,  perpendicular 
to  BC ;  then  in  AD  the  sine  of  B,  the  radius  being  BA 


Therefore,  take  BA,  and  make  it  a  parallel  distance  from 
90  to  90,  on  the  line  S;  then  take  AD,  and  moving  it 
along,  find  the  numbers  to  which  it  will  apply  as  a  pa- 
rallel distance 

At  a  given  point  A,  in  a  given  line  AB,  to  make  an 
angle  containing  any  number  of  degrees,  as  30. 

,C 


Make  AB  a  parallel  distance  from  90  to  90  on  S ;  take 
the  parallel  distance  from  30  to  30,  and  from  B  describe 
an  arc.  Draw  AC  touching  the  arc.  CAB  is  the  angle 
required. 

And  so  of  all  the  problems  under  the  protractor. 

6 


62 


A   TREATISE    ON    A    BOX    OP 


To  measure  any  angle,  ABC,  with  the  line  of  tangents,  T. 

,A 


At  any  distance,!),  raise  the  perpendicular  DE,  to  meet 
AB;  then  is  DE  the  tangent  of  B,  the  radius  being  BD. 
Make  BD  a  parallel  distance  from  45  to  45  on  T ;  then 
take  DE,  and  moving  it  along,  find  the  numbers  to  which 
it  will  apply  as  a  parallel  distance. 

At  a  given  point  A,  in  a  given  line  AB,  to  make  an 
angle  containing  any  number  of  degrees,  a,*  30 

D 


Take  any  point,  C,  at  which  erect  the  perpendicular  CD, 
Make  AC  a  parallel  distance  from  45  to  45.  Take  the 
parallel  distance  from  30  to  30,  and  from  C  cut  off  CE 
equal  to  it.  Join  AE.  EAB  is  the  angle  required. 

And  so  of  the  rest  under  the  protractor. 

To  measure  any  angle,  BAG,  with  the  line  of  secants. 


INSTRUMENTS   AND   \HE    SLIDE-RULE.  03 

'lake  any  point,  D,  and  raise  the  perpendicular  DE. 
Makc'AD  a  parallel  distance  from  0  to  0  on  the  line  of 
secants ;  then  take  AE,  and  moving  it  along,  find  the 
numbers  to  which  it  will  apply  as  a  parallel  distance. 

At  a  given  point,  A,  in  a  given  line  AB,  to  make  an 
angle  containing  any  number  of  degrees,  as  40 

C 


Kaise  BC  at  right  angles  to  AB.  Make  AB  a  parallel 
distance  from  0  to  0 ;  then  take  the  parallel  distance  from 
40  to  40,  and  with  it,  from  the  point  A,  cross  BC  in  D. 
Join  AD.  DAB  will  be  the  angle  required. 

And  so  of  the  rest  under  the  protractor. 

The  line  of  secants  cannot  be  employed  to  much  advantage 
when  the  number  of  degrees  is  under  30,  nor  the  line  of 
sines  when  above  70,  as  is  evident  from  an  inspection  of 
the  rule. 

To  find  the  chord,  sine,  tangent,  and  secant  of  30  de- 
grees, to  a  radius  of  two  inches,  AB.  (See  diagram  p.  64.) 
Take  2  inches  in  the  compasses,  and  make  it  a  parallel 
distance  from  60  to  60  on  the  scale  of  chords ;  it  will  also 
be  a  parallel  distance  from  45  to  45  on  the  tangents ;  and 
from  90  to  90  on  the  sines.  Therefore,  taking  the  parallel 
distance  from  30  to  30  on  the  line  C,  it  will  give  the  chord 


64 


A   TREATISE   ON   A   BOX   OF 


of  30  degrees,  BB;  from  30  to  30  on  the  line  S,  it  will 
be  the  sine  of  30  degrees,  BC ;  from  30  to  30  on  the  line 
T,  it  will  be  the  tangent  of  30  degrees,  BD.  When  the 
number  of  degrees  is  above  45,  the  same  opening  will  not 
6uffi.ce  for  the  tangents ;  it  will  then  be  necessary  to  set 
the  radius  to  the  45  of  the  smaller  tangents,  and  take  the 
aperture  as  usual.  So  for  the  secants,  make  2  inches  a 
parallel  distance  from  0  to  0 ;  then  will  the  parallel  dis- 
tance of  30  be  the  secant  AD. 

When  the  rule  is  opened  to  the  radius  of  the  smaller 
tangents,  it  is  also  opened  to  the  radius  of  the  secants. 

All  the  problems  given  under  the  protractor  may  be 
performed  by  any  of  these  lines ;  but  polygons  are  most 
conveniently  constructed  by  the  lines  POL.  ;  and  in  taking 
distances  from  these,  the  points  of  the  compasses  are  to  be 
placed  on  the  same  line  as  that  from  which  the  chords  are 
taken.  The  side  of  the  hexagon,  6,  does  not  reach  to  the 
chord  of  60,  as  has  been  mentioned;  a  smaller  radius 
being  chosen  for  the  purpose  of  including  the  5  and  4,  the 
pentagon  and  square.  Since  the  radius  of  a  circle  is  equal 
to  the  side  of  an  inscribed  hexagon,  the  radius  is  always 
to  be  made  a  parallel  distance  from  6  to  6. 

An  example  or  two  will  suffice. 


INSTRUMENTS   AND   THE    SLIDE-RULJ5. 


65 


To  construct  a  pentagon  on  a  given  line  AB.  Take 
AB,  and  make  it  a  parallel  distance  from  5  to  5.  Take 
the  parallel  distance  from  6  to  6  for  a  radius,  and  with  it 
from  A  and  B  describe  arcs  crossing  each  other  in  C,  and 
from  C  describe  a  circle.  The  distance  AB  will  run  5 
times  round  it,  and  form  the  pentagon  required. 


And  so  of  any  regular  polygon. 

In  a  given  circle,  to  inscribe  a  regular  heptagon.  Find 
the  centre  of  the  circle,  and  make  the  radius  a  parallel 
distance  from  6  to  6 ;  take  the  parallel  distance  from  7  to 
7,  and  it  will  run  7  times  round  the  circle  as  required. 


And  so  of  any  regular  polygon. 

6» 


66  A  TREATISE   ON   A   BOX    OF 


APPLICATION   OF   THE   SECTOR   TO   TRIGONOMETRY. 

llequired  the  height  of  a  tower,  AB,  the  angle  of  ele- 
vation, ACB,  200  feet  distant,  being  47 1°'. 


The  angle  B  will  be  90  —  47  £  =  42 }. 

Taking  AC  radius,  AB  is  the  tangent  of  C,  or  47  i°. 

Hence,  Rad.  :  CA  :  :  tang.  47 J°  :  AB. 

Take  200  from  a  scale  of  equal  parts,  (the  diagonal 
scale  is  the  best,)  and  make  it  a  parallel  distance  from  45 
to  45  on  the  smaller  line  of  tangents ;  take  the  parallel 
distance  from  47  J  to  47  i,  and  apply  it  to  the  same  scale 
of  equal  parts;  it  will  be  found  to  measure  about  218 
feet,  the  height  of  the  tower. 

Otherwise,  Sine  B  :  CA  :  :  sine  C  :  AB. 

On  the  line  of  sines  stretch  200  from  42  J  to  42  J ;  take 
the  parallel  distance  from  47  J  to  47J,  it  will  measure 
218  as  before. 

What  was  the  }v.;rpendicular  height  of  a  balloon,  when 
its  angles  of  elev;  I*  on  were  35°  and  64°,  C.K  taken  by  two 
observers  at  the  «iiiiu  time,  bc'n  on  the  rime  side  of  it, 


INSTRUMENTS  AND    THE    Sl.irE-RDLE. 


67 


and  in  the  same  vertical  line,  their  distance  beinp  880 
yards  ? 


The  external  angle  DEC  =  DAB  -f  ADK ; 
.-.  ADB  =  64  —  35^.29. 

Then,  Sine  ADB  :  AB  :  :  sine  A  :  BD. 

Take  880  from  a  scale  of  equal  parts,  and  make  it  a 
parallel  distance  from  29  to  29  on  the  line  of  sines ;  then 
will  the  parallel  distance  from  35  to  35  measure  1011  for 
the  line  BD. 

Again,  Sine  C  :  BD  :  :  sine  DEC  :  DC. 
Take  1041  and  make  it  a  parallel  distance  from  90  to 
90,  the  angle  C  being  a  right  angle ;  then  take  the  paral- 
lel distance  from  64  to  64,  and  it  will  measure  936  for 
CD,  the  perpendicular  height  of  the  balloon. 

Wanting  to  know  the  distance  between  two  inaccessible 
trees  from  the  top  of  a  tower,  120  feet  high,  which  lay 
in  the  same  right  line  with  the  two  objects  I  took  the 
angles  formed  by  the  perpendicular  wall  and  lines  con- 
ceived to  be  drawn  from  the  top  of  the  tower  to  the  bottom 
of  each  tree,  and  found  them  to  be  33°  and  64$°.  What 
1.4  the  «li<!tjiMA.c  th.  p  hi  t  «j»,'i  t»e  V:V* 


68 


A  TREATISE   ON   A  BOX  OF 


Angle,  BAG  =  33°.     Angle,  BAD  =  64 J. 
Making  AB  radius; — 

Rad.  :  AB  :  :  tang.  BAG  :  BC. 
Take  120  from  a  scale  of  equal  parts,  and  ma.ke  it  a 
parallel  distance  from  45  to  45  on  a  line  of  tangents ;  from 
which  take  the  parallel  distance  from  33  to  33 ;  it  will 
measure  78  on  the  scale  for  BC,  the  distance  of  the  first 
tree  from  the  bottom  of  the  tower. 

Again,  Rad.  :  AB  :  :  tang.  BAD  :  BD. 

Make  AB,  or  120,  a  parallel  distance  from  45  to  45  on 
the  smaller  line  of  tangents,  from  which  take  the  parallel 
distance  from  64£  to  64£;  it  will  measure  251  £  on  the 
scale  for  BD,  the  distance  of  the  farther  tree  from  the 
bottom  of  the  tower. 

Hence  251  £  —  78  =  173  J,  the  distance  between  the 
trees. 

Being  on  the  side  of  a  river,  and  wanting  to  know  the 
distance  to  a  house  which  was  seen  on  the  other  side,  I 
measured  200  yards  in  a  straight  line  by  the  side  of  the 
river;  and  then  at  each  end  took  the  horizontal  angle 
formed  between  the  house  and  the  other  end  of  the  Hue, 
which  were  68°  and  73°.  What  then  were  the  distances 
from  each  end  to  the  house  ? 


INSTRUMENTS   AND  THE   SLIDE-RULE.  09 


A  B 

Angle  A  =  68°  ;  B  =  73° ;  73°  and  68°  =  141  j 
.•.  C  =  39°,  since  the  three  anglesof  every  triangle  =  180° 
Sine  C  :  AB  :  :  sine  A  :  BC ;  and 
Sine  C  :  AB  :  :  sine  B   :  AC. 

Hence,  take  200  from  the  scale,  and  make  it  a  parallel 
distance  from  39  to  39  on  the  line  of  sines ;  then  will  the 
parallel  distance  from  68  to  68  measure  294  J  equal  parts 
for  BC;  and  the  parallel  distance  from  73  to  73  will 
measure  304  for  AC. 

Having  to  find  the  height  of  an  obelisk  standing  on  the 
top  of  a  declivity,  I  first  measured  from  its  bottom  a  dis- 
tance of  40  feet,  and  there  found  the  angle  formed  by  the 
oblique  plane,  and  a  line  imagined  to  go  to  the  top  of  the 
obelisk,  41°  ;  bat,  after  measuring  on  in  the  same  direc- 
tion 60  feet  farther,  the  like  angle  was  only  23°  45'. 
What  was  the  height  of  the  obelisk  ? 
B 


70 

AC  =  40;   DC  =  60;    angle  BAG  =41°;  BDC  = 
23t°j  . 


Then,  Sine  DEC  :  DC  :  :  sine  BDC  :  CB. 

Make  DC,  60  a  parallel  distance  on  the  line  of  sines 
from  171  t°  171;  take  the  parallel  distance  from  23f  to 
23  1,  and  it  will  measure  81  J  for  CB. 

Then,  in  the  triangle  ABC  ;  two  sides  BC,  CA,  being 
known,  and  their  included  angle, 

Sum  of  BC,  CA  :  diff.  BC,   CA  :  :  tang.  J  sum  of 
angles  A  and  B  :  tang  £  their  diff. 


C  +  CA  =  12Hj  and  BC  —  CA=4H. 

Again,  since  BCA  =  41°,  the  other  two  =  139°  ;  }  of 
which  =  69  £°  for  half  the  sum. 

Make  121J  a  parallel  distance  from  69J  to  69£  on  the 
line  of  tangents  ;  then  will  41  }  be  the  parallel  distance  of 
42  J.  Hence,  69  J  —  42i  =  27°  for  the  angle  CBA. 

Lastly,  Sine  CBA  :  CA  :  :  sine  BCA  :  AB. 

Make  40  a  parallel  distance  from  27  to  27  on  the  line 
of  sines;  then  will  the  parallel  distance  from  41  to  41 
measure  57  f  for  AB,  the  height  of  the  obelisk. 

Standing  upon  the  top  of  a  castle  80  feet  high,  I  threw 
a  string  over  to  a  person  on  the  farther  side  of  the  moat 
at  the  bottom,  and  found  it  to  measure  100  feet.  What 
was  the  breadth  of  the  moat,  and  angle  of  depression  ? 

Making  BC  radius  ;  BC  :  rad.      :  BA  :  sine  C. 
Make  100  a  parallel  distance  from  90  to  90  on  the  line 


INSTRUMENTS    AND   THE    SLIDE-RULE. 


S,  then  will  80  be  a  parallel  distance  from  53  to  53.  the 
angle  of  depression,  or  angle  C. 

Again,  Rad.  :  BC  :  :  cos.  C.  :  AC 

Letting  the  rule  stand,  the  aperture  is  set  to  the  first 
part  of  the  proportion  :  the  cosine  of  53°  is  37°.  Take 
the  parallel  distance  from  37  to  37,  and  apply  it  to  the 
scale  :  it  will  measure  60,  for  CA. 

The  breadth  of  a  street  is  30  feet,  the  height  of  a  house 
40.  What  must  be  the  length  of  a  ladder  that  will  reach 
from  the  top  of  the  house  to  the  opposite  side  of  the  way  ? 


A         so          B 

CB  +  BA  =  70;  CB— BA  =  10. 
J  sum  of  L  B,  C  and  A  =  45°. 

Hence  make  70  from  the  scale  a  parallel  distance  from 
45  to  45  on  the  larger  line  of  tangents  j  then  will  10  from 
the  same  scale  be  a  parallel  distance  from  8  to  8. 


72 


A  TREATISE   ON   A   BOX    OT 


8  +  45  =  53°  the  angle  CAB. 
Then,  Kad.  :  AB  :  :  sec.  CAB  :  AC. 

Make  30  a  parallel  distance  from  0  to  0  on  the  line  of 
secants;  then  will  the  parallel  distance  from  53  to  53 
measure  50  feet,  the  length  of  the  ladder  AC. 

Coasting  along  the  shore,  I  saw  a  cape  bear  from  me 
NNE;  then  I  stood  away  N  Wb  W  20  miles,  and  I  ob- 
served the  same  cape  to  bear  from  me  N  Eh  E :  required 
the  distance  of  the  ship  from  the  cape  at  her  last  station 


C,  the  ship's  first  station  ;  A,  the  place  of  the  ship  in 
ner  second  station  ;  B,  the  cape.     Then, 

CA=20  miles;  L 


=  78f  ;  L  ABC  = 

Hence,  Sine  ABC  :  AC  :  :  sine  ACB  :  AB. 

Make  20  a  parallel  distance  from  33f  to  33J  on  the 
line  of  sines;  then  will  the  parallel  distance  of  78f  measure 
35  miles  nearly,  for  AB,  the  distance  of  the  ship  from 
(he  cape  in  her  last  station 


INSTRUMENTS   AND   THE    SLIDE-RULE. 


73 


Enough  has  been  shown  to  enable  the  pupil  to  apply 
the  rule  to  most  of  the  purposes  to  which  it  is  adapted. 
Beside  the  sectoral  lines,  which  may  now  be  dismissed, 
there  are  on  the  edges  the  decimal  parts  of  a  foot,  and 
along  the  margin  a  scale  of  inches ;  these  need  no  expla- 
nation. There  are  also  three  other  lines,  marked  N,  S, 
and  T,  called  Gunter's  lines.  The  distances  on  N  are  the 
logarithms  of  numbers ;  on  S,  logarithmic  sines ;  and  on 
T,  logarithmic  tangents.  These  are  the  same  as  the  lines 
of  the  slide-rule. 


CONSTRUCTION. 


Referring  to  a  table  of  logarithmic  numbers,  sines,  and 
tangents,  we  have 


No. 

Log. 

o 

Sine. 

Tang. 

1 

.000 

1 

8.241 

8.241 

2 

.301 

2 

8.542 

8.543 

3 

.477 

3 

8.718 

8.719 

4 

.602 

4 

8.843 

8.844 

5 

.698 

5 

8.940 

8.941 

6 

.778 

6 

9.019 

9.021 

7 

.845 

&0. 

&o. 

&c. 

Lay  down  three  parallel  lines,  N,  S,  T,  as  below,  and 
draw  AB  perpendicular  to  them,  that  the  beginnings  of 
A 


j  a 


74  A   TREATISE   ON   A   BOX   OP 

the  three  may  correspond.  The  log.  of  1  being  0,  set  1 
at  the  commencement  N.  From  the  diagonal  scale  take 
off  301,  the  log.  of  2,  and  lay  it  from  N  to  2 ;  turn  it  over 
again,  and  it  will  mark  the  log.  of  4 ;  turning  it  again,  it 
will  reach  to  the  log.  of  8,  and  so  on. 

Again,  from  the  diagonal  scale  take  off  477,  the  log.  of 
3,  and  lay  it  from  N  to  3 ;  turn  it  over,  and  it  will  reach 
to  9,  from  9  to  27,  and  so  on.  Setting  the  same  distance 
from  2,  it  will  reach  to  6,  from  6  to  18,  and  so  on.  For 
5  take  698  from  the  scale,  and  set  it  from  N  to  5 ;  the 
same  distance  will  reach  from  2  to  10,  &c.  For  7  take 
845,  and  set  it  off  from  N  to  7.  Lay  the  line  over  again, 
and  proceed  to  fill  up  the  distances  11, 12,  &c.  from  a  set 
of  tables,  till  the  line  is  finished. 

For  the  line  S. 

The  logarithmic  sine  of  1°  is  8.241.  Disregarding  the 
whole  number  8,  which  is  prefixed  to  indicate  the  great 
extent  into  which  the  radius  is  supposed  to  be  divided, 
take  from  the  same  scale  241,  and  lay  it  from  S  to  1 ;  lay 
off  542  from  S  to  2 ;  718  from  S  to  3 ;  843  from  S  to  4; 
940  from  S  to  5 ;  1019  from  S  to  6;  and  so  on  to  90°. 

For  (lie  line  T. 

The  logarithmic  tangent  of  1°  is  8.241 ;  hence,  it  will 
be  over  the  1  of  S ;  lay  off  543  from  T  to  2 ;  719  from 
T  to  3 ;  844  from  T  to  4 ;  941  from  T  to  5 ;  1021  from 
T  to  6 ;  and  so  on  to  45.  Arriving  at  45,  the  numbers 
return,  and  50  is  placed  with  40 ;  30  with  60  ;  20  with 
70 ;  80  with  10 ;  and  90  at  the  beginning ;  the  decimal 
part  of  the  logarithmic  tangent  of  any  degree  being  the 


INSTRUMENTS   AND   THE   SLIDE-RULE.  75 

arithmetical  complement  of  the  cotangent;  since,  as  wag 
shown,  the  radius  is  a  mean  between  the  tangent  and  co 
tangent. 

USES. 

An  example  or  two  of  the  manner  of  using  these  lines 
will  be  sufficient.  For  the  reasons  of  the  operations,  the 
student  is  referred  to  the  Treatise  on  the  Slide-rule. 

In  the  lines  S  and  T,  the  numbers  show  the  values 
which  are  to  be  taken.  On  the  line  N,  the  second  division 
will  be  ten  times  those  of  the  first  division ;  the  values  are, 
otherwise,  arbitrary.  Thus,  if  3  of  the  first  division  be  3, 
that  of  the  second  will  be  30 ;  if  30,  300 ;  and  so  on. 

Uses  of  the  line  N,  or  Logarithmic  Numbers. 

To  multiply  4  by  5.  Take  the  distance  from  1  to  4, 
stretching  the  rule  open  to  its  full  extent ;  it  will  then 
reach  from  5  to  20,  the  product. 

To  divide  30  by  5.  Take  the  distance  from  1  to  5,  it 
will  reach  backwards  from  30  to  6,  the  quotient. 

To  find  a  fourth  proportional  to  three  numbers,  3,  4, 
aud  6.  That  is,  3  :  4  :  :  6  :  ?  Take  the  distance  from 
the  first  to  the  third ;  it  will  reach  from  the  second  to  the 
fourth.  The  distance  from  3  to  6  will  reach  from  4  to  8, 
the  fourth  proportional  required. 

To  square  and  cube  3.  Take  the  distance  from  1  to  3, 
and  set  it  forwards  from  3 ;  it  will  reach  to  9,  the  square 
required.  Set  it  forwards  again  from  9,  and  it  will  reach 
to  27,  the  cube. 

To  extract  the  square  root  and  cube  root  of  64.  Take 
tlir  distance  from  1  to  64;  find,  by  the  line  of  lines,  the 
half  of  this;  it  will  roach  from  1  to  8,  the  square  root 


76 


A   TREATISE   ON   A   BOX   OF 


Find  by  the  line  of  lines  the  third  of  it,  and  it  will  reach 
from  1  to  4,  the  cube  root  of  64. 

Uses  of  the  lines  N,  S,  and  T,  conjointly. 

In  the  right-angled  triangle  ABC,  suppose  AB  124, 
and  the  angle  A  34°  20';   consequently,   the  angle  0 

55°  40'.     Required  BC  and  CA. 

C 


124 

Sine  C  :  sine  A  :  :  AB  :  BC. 

Hence,  on  the  line  S,  take  the  distance  backwards  from 
55|  to  34£ ;  this  will  reach  back  on  the  line  N  from  124 
to  84.7  the  length  of  BC. 

Again,  Sine  C  :  sine  B  :  :  AB  :  AC. 

Take  the  distance  forwards  on  the  line  S  from  55f  to 
90;  this  will  reach  forwards  on  the  line  N  from  124  to 
150,  the  length  of  AC. 

Otherwise  for  the  perpendicular  BC, 

Bad.  :  tang.  A  :  :  AB  :  BC. 

On  the  line  T,  take  the  backward  distance  from  46  to 
34£ ;  this  will  reach  back,  on  the  line  N,  from  124  to 
84.7  for  BC,  as  before. 

From  the  top  of  a  tower,  by  the  sea-side,  of  143  feet 


INSTRUMENTS   AND    THE    SLIDE-RULE.  I  i 

height,  it  was  observed  that  the  angle  of  depression  of  a 
ship's  bottom,  then  at  anchor,  measured  35°  :  what  then 
was  the  ship's  distance  from  the  bottom  of  the  wall  ? 


The  angle  of  depression  of  the  vessel  is  ABC,  and  con- 
sequently is  equal  to  the  angle  of  elevation  of  the  tower, 
BCD.  Hence,  making  BD  radius ; 

Had.  :  tang.  55°  :  :  BD  :  DC. 

Stretch  the  compasses  on  the  line  T,  from  45  to  55; 
this  will  reach  from  143  to  204  on  the  line  N. 

What  is  the  perpendicular  height  of  a  hill ;  its  angle  of 
elevation,  taken  at  the  bottom  of  it,  being  43°;  and  200 
yards  farther  off,  on  a  level  with  the  bottom  of  it,  31°? 


A  200  B 


78  A   TREATISE  ON   A   BOX   OP 

Sine  15°  :  sine  31°  :  :  200  :  BC;  and 
Sine  90     :  sine  46°  :  :  BC  :  CD. 

Stretch  the  compasses  on  the  line  S  from  15  to  31 ;  this 
distance,  on  the  line  N,  will  reach  from  200  to  398  for 
BC.  Again,  take  the  backward  distance  from  90  to  46 
on  S;  this  will  reach  back  from  398  to  286  for  CD,  the 
height  of  the  hill. 

A  point  of  land  was  observed,  by  a  ship  at  sea,  to  bear 
E  b  S;  and  after  sailing  N  E  12  miles,  it  was  found  to 
bear  S  E  b  E.  It  is  required  to  determine  the  place  of 
that  headland,  and  the  ship's  distance  from  it  at  the  last 
observation. 


Sine  22J  :  sine  56J  :  :  12  :  BD. 

On  the  line  S  take  the  forward  distance  from  22  i  to 
56  J ;  this  will  reach  from  12  to  26  for  BD,  the  ship's 
distance  from  the  last  place  of  observation. 

Standing  upon  the  top  of  a  castle  80  feet  high,  I  threw 
,.  string  over  to  a  person  on  the  farther  side  of  the  moat 


INSTRUMENTS   AND   THE   SLIDE-RULE. 


79 


at  bottom,  and  found  it  to  measure  100  feet.     Required 
the  breadth  of  the  moat,  and  the  angle  of  depression. 


Take  CB  radius ;  then — 

CB  :  BA      :  :  rad.  :  sine  C ;  and 
Rad.  :  cos.  C  :  :  CB   :  CA. 

Measure  back  on  N  from  100  to  80 ;  this  on  S  will 
reach  back  from  90  to  53  for  the  sine.  The  cosine  of 
53  to  37. 

On  S  take  the  distance  back  from  90  to  37 ;  it  will 
reach  back  on  N,  from  100  to  60  for  CA. 

These  examples  will  be  sufficient  to  show  the  pupil  the 
method  of  using  the  lines.  He  can  work  over  the  ques- 
tions that  were  solved  by  the  natural  sines  and  tangents 
of  the  sectoral  lines ;  taking  the  numbers  from  N,  instead 
of  from  a  scale  of  equal  parts,  and  the  sines  and  tangents 
from  S  and  T  respectively;  observing,  that  a  forward 
distance  from  one  is  to  be  applied  as  a  forward  distance 
to  the  other,  and  a  backward  distance  from  one  as  a  back- 
ward distance  to  the  other.  The  distance  on  the  line  T, 
from  45  to  55,  is  to  be  reckoned  as  a  forward  distance, 
while  the  distance  from  45  to  35,  though  the  same,  is  to 
be  accounted  a  backward  distance.  He  will  also  observe, 
in  stating  the  proportion,  that  that  which  was  made  the 


80  A  TREATISE   ON   A  BOX   OF 

second  term  for  the  sectoral  lines,  must  be  placed  third 
when  intended  for  the  Gunter's.     Thus — 

For  the  sectoral,        AB  :  sine  C  :  :     BC     :  sine  A. 

For  the  logarithmic,  AB  :     BG     :  :  sine  C  :  sine  A. 

Such  is  the  Sector;  it  was  devised  by  the  celebrated 
Gunter,  about  the  year  1607,  and  did  not,  at  first,  con- 
tain the  N,  S,  and  T  lines;  these  were  added  sixteen 
years  later,  or  nine  years  after  Napier's  admirable  inven- 
tion of  logarithms.  In  the  year  1657,  Partridge  greatly 
improved  upon  the  plan,  by  laying  the  lines  down  double, 
and  sliding  one  against  the  other.  The  sector  dwindles 
into  insignificance,  in  comparison  with  the  slide-rule, 
which  is  nearly  a  perfect  instrument,  and  adapted,  in  a 
degree,  to  every  species  of  computation  for  which  loga- 
rithms are  available.  The  chief  merit,  however,  is  due 
to  Gunter,  for  hundreds  can  improve  where  only  one  can 
invent. 


INSTRUMENTS   AND   THE    SLIDE-RULE.  81 


LOGARITHMS. 

LOGARITHMS  are  a  series  of  numbers  in  arithmetical 
progression,  corresponding  to  another  series  of  numbers 
in  geometrical  progression  :  thus — 

0123456  7  8 

1     2    4    8     16    3'2    64     128    256 

where  the  indices  0, 1,  2,  3,  &c.  in  the  arithmetical  series 
are  the  logarithms  of  the  numbers  1,  2,  4,  8,  &c.  of  the 
geometrical. 

On  examining  these,  it  will  be  found  that  if  any  two 
of  the  logarithms,  or  indices,  are  added  together,  their 
sum  will  be  the  logarithm  or  index  of  the  product  of  the 
numbers  to  which  they  belong.  Thus,  2  and  3  are  5  j 
the  number  against  this  is  32,  which  is  the  product  of  4 
and  8,  the  numbers  beneath  the  indices  2  and  3. 

In  like  manner,  if  any  one  of  the  indices  be  subtracted 
from  another,  their  difference  is  the  index  of  the  quotient 
of  the  numbers.  Thus,  5  from  7  leave  2,  the  number 
against  which  is  4,  the  quotient  of  128  by  32. 

For  the  same  reason,  if  any  one  of  the  indices  be  mul- 
tiplied by  another  denoting  power,  the  product  will  be 
the  index  of  that  power.  Thus,  to  find  the  square  of  8 ; 
its  index  is  3,  which,  doubled,  becomes  6;  the  index  of 
6-t,  the  square  of  8,  as  required. 

Lastly,  if  the  index  of  any  number  be  divided  by  the 
index  of  any  root,  the  quotient  will  be  the  index  of  that 


82  A   TREATISE   ON   A  BOX   OF 

root.  Thus,  to  find  the  square  root  of  16 ;  its  index  is  4, 
the  half  of  which  is  2,  which  is  the  index  of  4,  the  square 
root  of  16,  as  required.  From  which  it  appears  that  ad- 
dition of  logarithms  answers  to  multiplication  of  common 
numbers,  subtraction  to  division,  multiplication  to  invo- 
lution, and  division  to  evolution.  The  same  will  also  be 
the  case  if  we  select  any  other  geometrical  series ;  thus — 

0123  4  5          6 

1    3     9    27     81     243  729 
or — 

i   10   100   1,000   10,000   ioo,5ooo   1,000,000 

From  which  it  is  evident  that  the  same  indices  may  serve 
for  any  system,  and,  consequently,  that  the  varieties  of 
systems  of  logarithms  are  infinite. 

Now,  since  numbers  are  but  expressions  for  the  measure 
of  distances  from  a  fixed  point,  it  follows,  that  if  from  the 
commencement  0  of  any  line,  we  set  off  equal  distances  in 
the  points  1,  2,  3,  &c.,  and  raise  against  them  a  series  of 
perpendiculars,  1,  2,  4,  8,  &c.,  we  shall  have  in  the  extre- 
mities of  these  perpendiculars  a  series  of  distances  1,  2,  4, 
8,  &c.,  whose  logarithms  will  be  the  distances  0, 1,  2,  3,&c. 
These  representing  the  indices  of  the  distances  measured 
by  the  perpendiculars,  they  will  of  course  possess  the  same 
properties  as  the  indices  themselves. 

Thus,  let  it  be  required  to  multiply  16  by  4.  With  a 
pair  of  compasses  take  the  distance  from  0  to  2,  the  index 
corresponding  to  4 ;  set  one  foot  of  the  compasses  on  4, 
the  index  of  16,  the  other  point  will  reach  forwards  to  6, 
the  index  of  64,  the  product  of  the  numbers  4  and  16. 

'Again,  let  it  be  required  to  divide  64  by  16.  Take  the 
distance  from  0  to  4,  the  index  of  the  less;  place  one  foot 


INSTRUMENTS   AND   THE   SLIDE-RULE.  83 

GT 


•\S 


«_LL 


SC 


0123456  01        2        3        4        5        6 

of  the  compasses  on  6,  the  index  of  the  greater,  the  other 
point  will  reach  lack  to  2,  the  index  of  4,  the  quotient 
required.  Next,  let  it  be  required  to  find  the  square  of  8. 
Take  the  distance  from  0  to  3,  the  index  of  8.  Place  the 
compasses  on  the  point  0,  and  turn  them  over  twice;  they 
will  reach  to  6,  the  index  of  64,  the  square  required. 

Lastly,  to  find  the  square  root  of  64.  Take  half  the 
distance  from  0  to  6,  which  is  the  point  3,  the  index  of  8, 
the  square  root  required. 

But  for  taking  squares,  and  square  roots,  it  will  be  more 
convenient  to  have  the  logarithmic  distances  laid  down  to 
two  scales,  Figs.  1  and  2,  in  which  the  distances  of  the 
latter  shall  be  twice  those  of  the  former.  Then  to  take 
the  square  of  8.  Measure  the  distance  from  0  to  3, 
Fit/.  2  ;  apply  this  to  F'KJ.  1 ;  and  it  will  reach  to  6,  tho- 
index  of  64,  the  square  required.  And  to  find  the  square 
root  of  64.  Take  the  distance  from  0  to  6,  F'uj.  lj  it  will 


84  A   TREATISE   ON   A   BOX    OF 

reach  from  0  to  3,  Fig.  2,  which  is  the  index  of  8,  the 
square  root  required.  Thus,  by  mechanical  means,  we 
obtain  the  same  results  as  by  arithmetical  calculation  :  a 
forward  motion  performs  the  work  of  multiplication ;  a 
backward,  that  of  subtraction ;  an  increase  of  distance,  the 
raising  of  powers;  and  a  diminution,  the  extraction  of 
roots. 

But  since  the  application  of  the  compasses  is  a  tedious 
method,  it  is  desirable  to  perform  the  same  operations  by 
a  readier  way,  which  we  may  now  proceed  to  consider. 

If  to  the  end  of  AB  we  set  the  beginning  of  CD,  it  is 
evident  that  the  distance  AE,  reaching  from  the  com- 
mencement of  one  to  the  end  of  the  other,  measures  their 
united  lengths,  or  expresses  the  sum  of  the  two  ; 


That  is,  AB  +  CD  =  AE  =  2  +  3  =  5. 

And  if  against  E,  the  extremity  of  the  line  AE,  we  set 
D,  the  extremity  of  CD;  the  part  AB,  between  the  be- 
ginning of  the  longer  and  the  beginning  of  the  shorter, 
measures  their  difference ; 

That  is,  AE  —  CD  =  AB  =  5—  3  =  2. 

For  addition,  then,  the  rule  may  be  expressed : — Set  the 
beginning  of  one  to  the  end  of  the  other ;  then  against  the 
end  of  the  second  is  their  sum  on  the  first.  And,  for  sub- 
traction : — Set  the  ends  togetlier,  then  at  the  beginning  of 
the  shorter  is  their  difference  on  the  longer. 

A  few  operations  will  best  imprint  this  upon  the  me- 
mory of  the  student,  for  which  purpose  he  must  furnish 
himself  with  scales  of  equal  parts,  as  on  p.  85. 


INSTRUMENTS   AND   THE    SLIDE-RULE. 


85 


On  a  sheet  of  paper  take  any  distance,  and  set  it  off  20 
times,  as  A.     And  beneath  it  at  any  convenient  distance, 

Common  Scale. 


J'D 


\    2  3  4   jSf  fr    8 


J? 


Stofe. 

double  the  space  10  times,  as  D.  Then  take  a  slip  of 
card  or  paper,  B,  which  we  may  call  the  slide,  of  a  breadth 
about  sufficient  to  fill  up  the  space  left  between.  A  and  D, 
and  mark  upon  it  the  same  scale  of  equal  parts  as  upon 
A.  With  these  we  may  perform  the  following  opera- 
tions :  — 

I.  To  find  the  half  of  any  number,  and  the  double  uf 
any  number. 

Lay  the  slide  in  the  space  between  A  and  D,  then  under 
any  number  on  B  is  its  half  on  D,  and  over  any  number 
on  D  is  its  double  on  B. 

II.  To  find  the  sum  of  any  two  numbers,  and  half  theii 
gum. 

Let  the  numbers  be  6  and  4.  Against  the  6  on  A  set 
the  beginning  on  B,  then  over  the  4  on  B  is  10,  their 
sum,  and  under  it  on  D  is  5,  half  their  sum. 

Ill  To  find  the  difference  of  any  two  numbers,  and 
half  their  difference. 

8 


86  A   TREATISE   ON   A   BOX   OP 

Let  the  numbers  be  11  and  5.  Against  the  11  on  A 
set  the  5  on  B,  then  over  the  beginning  of  B  is  6,  then 
difference,  and  under  it  3,  half  their  difference. 

IV.  The  sum  of  two  numbers  may  be  found  by  three 
methods. 

Thus,  of  5  and  7.  Against  5  on  A  set  the  beginning 
of  B,  over  the  7  on  B  is  12  on  A; — or  push  the  slide  out 
till  the  5  on  B  reach  the  beginning  of  A,  then  under  the 
7  on  A  is  12  on  B ; — or  invert  the  slide,  and  set  the  7  on 
B  to  the  5  on  A,  then  against  the  beginning  of  B  is  12  on 
A.  Also  for  the  difference ;  under  the  7  on  A  set  the  5 
on  B,  against  the  beginning  of  B  is  2  on  A ; — or  against 
the  5  on  A  set  the  7  of  B,  against  the  beginning  of  A  is 
2  on  B; — or  invert  the  slide  and  set  its  beginning  to  the 
7  of  A,  then  over  the  5  of  B  is  2  on  A. 

V.  To  add  any  number,  3,  to  twice  another,  4. 

Push  the  slide  back  till  the  3  on  B  reaches  the  begin- 
ning of  D  ;  over  the  4  on  D  is  11  on  B.  Or  against  the 
4  on  D  set  the  beginning  of  the  slide  :  over  the  3  of  this 
is  1 1  on  A  — Hence,  to  multiply  any  number  by  3,  as  6 ; 
(that  is,  to  add  6  to  twice  6.)  To  the  6  on  D  set  the  6 
on  the  slide  inverted;  over  the  beginning  of  this  is  18  on 
A. — Hence  also  to  divide  any  number  by  3,  as  18.  Set 
the  beginning  of  the  slide  inverted  to  18  on  A,  or  the  18 
on  B  to  the  beginning  of  A  or  D,  then  where  equal  values 
meet  together  on  B  and  D,  is  the  third  of  the  number; 
thus,  the  6  on  B  meets  the  6  on  D. 

VI.  To  add  a  number,  5,  to  the  half  of  another,  6. 
Against  5  on  D  set  the  beginning  of  B,  under  the  6  of 

which  is  8  on  D. 


INSTRUMENTS   AND   THE    SLIDE-RULE.  87 

VII.  To  subtract  a  number,  3,  from  twice  another,  4. 
Against  the  4  on  D  set  the  3  of  B,  over  the  beginning 

of  which  is  5  on  A. 

VIII.  To  subtract  half  a  number,  6,  from  another,  7. 
Against  the  7  on  D  set  the  6  of  B ;  under  the  begin- 
ning ol  this  is  4  on  D. 

IX.  From  any  number,  9,  to  subtract  twice  another,  2. 
Against  9  on  B  set  the  2  of  D ;  over  the  beginning  of 

D  is  5  on  B  :  or  invert  the  slide,  and  over  the  beginning 
of  D  set  9  on  B ;  over  the  2  of  D  is  5  on  B. 

X.  To  subtract  a  number,  3,  from  the  sum  of  two 
others,  4  and  5. 

Against  the  3  on  A  set  the  4  on  B ;  then  under  the  5 
of  A  is  6  on  B.  The  reason  of  this  is  plain  ;  for  to  have 
added  4  to  5,  the  slide  ought  to  have  been  pushed  out  till 
the  4  fell  under  the  beginning  of  A ;  but,  as  it  was  not 
removed  so  far  back  by  3  spaces,  the  result  will  evidently 
be  3  less  than  the  sum.  Otherwise,  invert  the  slide,  and 
against  the  4  on  A  set  the  5  of  B ;  over  the  3  of  this  is 
6  on  A,  or  under  the  3  of  A  is  6  on  B.  Here,  had  we 
wanted  the  sum,  we  should  have  counted  to  the  beginning 
of  the  slide,  or  from  the  beginning  of  A ;  but  as  in  both 
instances  we  omitted  3  spaces,  the  result  is  3  less  than  the 
sum. 

XI.  To  subtract  3  from  5  added  to  twice  4. 

Under  the  3  of  A  set  the  5  of  B  ;  over  the  4  of  D  is  10 
on  B.  The  reason  of  this  is  obvious  :  to  have  added  5  to 
twice  4,  the  slide  ought  to  have  been  pushed  out  till  the  5 
reached  the  beginning  of  A  or  D,  but  as  it  was  not  re- 


88  A   TREATISE    ON   A    BOX   OP 

moved  so  far  by  3  spaces,  the  result  is  3  less  than  the 
sum.  It  may  be  performed  otherwise  by  inverting  the 
slide.  Against  the  4  of  D  set  the  5  of  B ;  over  the  3  of 
this  is  10  on  A,  or  under  the  3  of  A  is  10  on  B.  The 
reason  is  evident. 

XII.  To  subtract  twice  a  number,  3,  from  the  sum  of  5 
and  twice  4. 

Place  the  5  of  B  over  the  3  of  D ;  over  the  4  of  D  is  7 
on  B.  To  have  added  5  to  twice  4,  the  5  of  B  ought  to 
have  been  set  at  the  beginning  of  D,  but  as  it  is  not  re- 
moved so  far  back  by  twice  3  spaces,  the  result  is  twice  3 
less  than  the  sum. — These  operations  are  to  be  carefully 
attended  to,  especially  the  last,  as,  in  working  with  the 
slide-rule,  it  is  more  employed  than  any  other. 

XIII.  To  substract  half  6  from  5  added  to  half  8. 
Against  the  5  on  D  set  the  6  of  B ;  under  the  8  of  B 

is  6  on  D.  To  have  added  5  to  half  8,  the  beginning  of 
the  slide  ought  to  have  been  placed  at  the  5  of  D,  then 
under  the  8  of  B  would  have  been  the  sum ;  but  as  the 
slide  is  not  set  so  forward  by  6  half-spaces,  the  result  is 
half  6  less  than  the  sum. 

As  the  whole  art  of  using  the  slide-rule  depends  upon  a 
perfect  knowledge  of  these  simple  movements,  the  pupil 
will  do  well  to  make  himself  thoroughly  acquainted  with 
them,  and  to  attend  carefully  to  the  reasons  for  every  ope- 
ration. Unless  he  does  this,  he  must  always  work  in  the 
dark,  and  will  be  perpetually  liable  to  fall  into  mistakes ; 
whereas,  if  he  makes  himself  intimate  with  them,  he  will 
be  enabled  to  proceed  with  certainty  and  pleasure. 

We  may  now  resume  the  consideration  or  logarithms, 


INSTRUMENTS   AND   THE    SLIDE-RULE. 


89 


and  return  to  FIGS.  1  and  2,  on  page  83 ;  and  here  it  is 
at  once  evident,  that  as  far  as  the  solution  of  any  questions 
is  concerned,  the  perpendiculars  are  of  no  importance,  the 
equal  distances  and  the  increasing  numbers  being  all  that 
are  required.  If,  then,  to  the  scales  we  have  been  using, 
or  to  any  others  of  equal  parts,  where  the  distances  on  D 
are  double  those  of  A  and  B,  we  affix  the  geometrical 
numbers  1,  2,  4,  8,  &c.,  the  distances,  measured  from 
the  commencement,  will  be  the  logarithms  of  those  num- 
bers, »nd  may  be  applied  to  the  usual  purposes  for  which 
logarithms  are  adapted. 

The  preceding  operations  may  now  be  repeated ;  but, 
instead  of  simply  adding  and  subtracting,  doubling  and 
halving,  they  will  present  themselves  under  the  shapes  of 
multiplying  and  dividing,  squaring  and  extracting  the 

Logarithmic  Scale. 


I      2       4      s      if     3i? 

Cjl     l^S    2^6'    512     ID' 

21  A 
2    D 

1 

1^4 

1        i 

J              3 

/: 


128 


SJ2 


Slide. 

square  root ;  and  will  become  converted  into  the  following 
operations,  which  the  pupil  should  compare  with  the  pre- 
ceding, step  by  step,  as  he  advances,  making  use  of  the 
logarithmic  scale  and  slide,  and  the  common  scale  and 
elide,  alternately. 


90  A  TREATISE  ON   A  BOX   OF 

I.  To  find  the  square  of  any  number,  and  the  square 
root  of  any  number. 

Lay  the  slide  in  the  space  between  A  and  D ;  then 
under  any  number  on  B  is  its  square  root  on  D ;  and  over 
any  number  on  D  is  its  square  on  B. 

II.  To  find  the  product  of  any  two  numbers,  4  and  16, 
and  the  square  root  of  their  product. 

To  4  on  A  set  the  beginning  of  B,  over  the  16  of  which 
is  64,  their  product  on  A ;  and  under  it,  8,  the  square  root 
of  their  product,  the  mean  proportional  between  them. 

III.  To  find  the  quotient  of  any  two  numbers,  8  and 
128,  and  the  square  root  of  their  quotient. 

Against  the  128  on  A  set  the  8  on  B ;  then  over  the 
beginning  of  B  is  16,  their  quotient;  and  under  it,  4,  the 
square  root  of  their  quotient. 

IV.  The  product  of  two  numbers  may  be  found  by  three 
methods. 

Thus,  of  16  and  4.  Against  16  on  A  set  the  begin- 
ning of  B ;  over  the  4  of  B  is  64  on  A; — or,  push  the 
slide  out  till  the  4  on  B  reaches  the  beginning  of  A ; 
then  under  the  16  of  A  is  64  on  B ; — or,  invert  the  slide, 
and  set  the  4  on  B  to  the  16  on  A;  then  against  the  be- 
ginning of  B  is  64  on  A.  Also,  for  the  quotient : — under 
the  16  on  A  set  the  4  on  B ;  against  the  beginning  of  B 
is  4  on  A; — or,  against  the  4  on  A  set  the  16  of  B; 
against  the  beginning  of  A  is  4  on  B ; — or  invert  the 
slide  and  set  its  beginning  to  the  16  of  A ;  then  over  the 
4  of  B  is  4  on  A. 

V.  To  multiply  any  number,  4,   by  the  square  of 
another,  8. 


INSTRUMENTS    AND   THE    SLIDE-RULE.  91 

Push  the  slide  back  till  the  4  on  B  reaches  the  begin- 
ning of  D ;  over  the  8  on  D  is  256  on  B ;  or  against  the 
8  on  D  set  the  beginning  of  the  slide ;  over  the  4  of  this 
10  256  on  A. 

Hence,  to  cube  any  number,  as  8,  that  is,  to  multiply 
8  by  the  square  of  8.  Against  the  beginning  of  D  set  8 
on  B ;  over  the  8  of  D  is  512  on  B  ; — or,  invert  the  slide, 
and  against  8  on  D  set  8  on  B ;  over  the  beginning  of  B 
is  512  on  A. 

Hence,  to  find  the  cube  root  of  a  number,  as  512.  Set 
the  beginning  of  the  slide  inverted  to  512  on  A,  or  512 
on  B  to  the  beginning  of  A  or  D ;  then  where  equal 
values  meet  together  on  B  and  D  is  the  cube  root  of  the 
number,  8. 

VI.  To  multiply  a  number,  8,  by  the  square  root  of 
another,  16. 

Against  the  8  on  D  set  the  beginning  of  B,  under  the 
16  of  which  is  32  on  D. 

VII.  To  divide  the  square  of  any  number,  16,  by  any 
number,  8. 

Against  the  16  on  D  set  the  8  of  B,  over  the  beginning 
of  which  is  32  on  A. 

VIII.  To  divide  any  number,  64,  by  the  square  root 
of  any  number,  16. 

Against  the  64  on  D  set  the  16  of  B ;  under  the  begin- 
ning of  this  is  16  on  D. 

IX.  To  divide  any  number,  64,  by  the  square  of  another,  4. 
Against  64  on  B  set  the  4  of  D;  over  the  beginning  of 

D  is  4  on  B. 


92  A  TREATISE   ON   A  BOX   OF 

X.  To  find  a  fourth  proportional  to  three  numbers.  4, 
82,  and  64;  or,  which  is  the  same  thing,  to  divide  by  4, 
32  times  64. 

Against  the  4  on  A  set  the  32  on  B ;  then  under  the 
64  on  A  is  512  on  B.  For  inverse  proportion, — as,  if  8 
men  can  build  a  wall  in  16  days,  in  how  many  days  will 
32  perform  the  same  ? — Invert  the  slide,  and  against  the 
8  on  A  set  the  16  of  B ;  under  the  32  of  this  is  4  on  B; 
or,  over  the  32  of  B  is  4  on  A. 

XI.  To  divide  by  8,  32  times  the  square  of  4. 
Under  the  8  of  A  set  the  32  of  B ;  over  the  4  of  D  is 

64  on  B  ; — or,  invert  the  slide,  and  against  the  4  of  D  set 
the  32  of  B  j  over  the  8  of  this  is  64  on  A,  or  under  the 
8  of  A  is  64  on  B. 

XII.  To  divide  by  the  square  of  4,  8  times  the  square 
of  16. 

Place  the  8  of  B  over  the  4  of  D ;  over  the  16  of  D  is 
128  on  B. 

XIII.  To  divide  by  the  square-root  of  64,  4  times  the 
square  root  of  256. 

Against  the  4  of  D  set  the  64  of  B;  under  the  256  of 
B  is  8  on  D. 

These  operations,  it  will  be  seen  at  once,  are  precisely 
the  same  as  the  former.  They  may  be  represented  alge- 
braically, as  beneath;  where  m,  n,  and  r,  are  any  num- 
bers taken  at  pleasure. 

I  Log-  m*  =  2  log.  m ; 

.*.  m  on  D,  m2  on  B. 
Log.  -j/  m  =  £  log.  m ; 
•.  M  on  B,  j/m  on  D. 


INSTRUMENTS   AND  THE   SLIDE-RULE.  93 

IL  Log.  m  n  =  log.  m  -f-  log.  n ; 

.-.  m  on  A  -f-  n  on  B;  m  n  on  A 

log,  m  -4-  log,  n 

Jx)g.  V  »* » =-  2  ' 

.-.  m  on  A  -j-  n  on  B,  1/w  n  oil  D. 


IH  Log.  —  =  log.  m  —  log.  n  ; 


M 

.-.  TO  on  A  —  n  on  B,  —  on  A. 
n 

Log.    /?»_log.  m.  —  log,  n; 
\  n~  2 

.•.  TO  on  A  —  n  on  B,A  /!!ion  D 


IV.  Log.  TO  n  =  log.  TO  -f-  log.  « ; 
.•.  TO  on  A  -{-  n  on  B,  TO  »  on  A 
Or,  TO  on  B  -J-  n  on  A,  TO  n  on  B, 

and  log.  -=  log.  m.  — log.  »; 

fir 

TO 

.•.  TO  on  A  —  n  on  B,  —  on  A 
Or,  TO  on  B  =n  on  A,  — on  B. 

V.  Log.  TO  n9  =  log.  m  -f  2  log.  n  ; 
.-. »»  on  B  -J-  n  on  D,  m  n*  on  B. 


4  A    TREATISE   ON   A   BOX   OF 

Or,  72  on  D  -{-  m  on  B,  m  n2  on  A, 

and  log.  m3  or  m  ma  =  log.  m  -J-  2  log.  m  j 

.•.  m  on  B  -}-  m  on  D,  m3  on  B. 

Or,  m  on  D  -j-  m  on  B,  m3  on  A. 

log.  71 

VI.  Log.  m  7i  J  =  log.  m  -] — -| — ; 

.-.  m  on  D  -f-  7<  on  B,  m  ?i  J  on  D. 

m3 

VII.  Log.  —  =  2  log.  m  —  log.  n ; 

n 

.-.  m  on  D  —  n  on  B.  —  on  A. 

n 

VIII.  Log  ™=log.m-!^; 

7i2"  2t 

fTL 

.•.  m  on  D  —  n  on  B,  —  on  D. 

IX.  Log.  —  =  log.  m  —  2  log.  n : 

na 

.-.  in  on  B  —  n  on  D,  —  on  B. 

X.  Log.  —  —  =  log.  m  -{-  log.  n  —  log.  r; 
.•.  —  r  on  A  -j-  m  on  B  -}-  n  on  A, on  B. 

XI.  Log.  -  —  =  log.  m  -{-  2  log.  n  —  log.  r; 

.•.  — ;  on  A  +  w  on  B  +  n  on  D. on  B. 

r 


INSTRUMENTS  AND   THE   SLIDE-RULE. 


XII.  Log.  —  —  =  log.  m  -j-  2  log.  n  —  2  log.  r; 


...  —  r  on  D  -j-  m  on  B  -f-  n  on  D, 


m  n8 


on  B. 


XIII.    Log. 


,   log.  n      log.  r 
=  log.m+-| 1-; 


•.  —  r  on  B  -f-  m  on  D  -\-  n  on  B, 


on  D. 


In  all  these  operations  the  student  will  at  once  perceive, 
what  it  is  scarcely  necessary  to  mention,  that  the  move- 
ments are  from  the  paper  to  the  slide,  and  from  the  slide 
to  the  paper,  alternately. 

Thus,  from  A  to  B,  and  back  to  A.  From  A  to  B,  and 
thence  to  D.  From  B  to  D,  and  back  to  B.  From  D  to 
B,  and  back  to  D.  From  D  to  B,  and  thence  to  A.  From 
A"  to  B,  and  thence  to  A,  and  back  to  B.  From  A  to  B, 
and  thence  to  D,  and  back  to  B. 

The  preceding  operations  include  the  whole  theory  of 
the  Slide-rule;  but  as  it  is  suitable  only  for  particular 
numbers,  in  the  form  we  have  presented  it,  it  remains  to 
explain  the  method  of  inserting  those  that  have  been 
omitted.  For  this  purpose,  draw  any  angle  BAG,  and  in 


F    H      L 


the  base  take  any  two  points,  D,  F ;  make  AE  =  AF ; 
join  DE,  EF,  and  through  F  draw  FG  parallel  to  DE ; 


90 


A  TREATISE   ON   A   BOX   OF 


and  through  G  draw  GH  parallel  to  FE;  and  so  on. 
Then  we  have, 

AD  :  AE  :  :  AF  :  AG.     But,  AE  =  AF; 
.-.AD  :  AF  :  :  AF  :  AG.     But,  AG=AH; 

.-.AD  :  AF  :  :  AF  :  AH. 
Hence,  AD  :  AF  :  :  AF  :  AH  :  :  AH  :  AL,  &c. 

Putting  AD  =  1,  and  AF  =  a,  we  have 

1  :  a  :  :  a  :  aa  :  :  az  :  a3  :  :  a3  :  a4,  &c.; 
or,  since  a°  =  1, 

a°  :  a1  :  :  a1  :  a9  :  :  a3  :  a3  :  :  a3  :  a4,  &c. ; 

where  the  indices,  0,  1,  2,  3,  4,  &c.  are  the  logarithms 
of  a°,  a1,  aa,  a3,  a4,  &c.,  or  of  AD,  AF,  AH,  AL,  &c. 

Along  any  line,  MN,  from  any  point  0,  set  off  a  num- 
ber of  equal  distances,  1,  2,  3,  4,  &c.,  and  at  these  erect 
perpendiculars,  taking  the  first  equal  to  AD ;  the  next, 
AF ;  the  next,  AH ;  and  so  on. 


M 


-N 


Then  shall  0  be  the  logarithm  of  AD  or  1;  01  the 
logarithm  of  AF;  0  2  of  AH  j  0  3  of  AL  j  and  so  on. 

If  the  distance  between  the  points  0,  1,  2,  3,  &c.  were 
indefinitely  small,  then  would  the  line  RS,  connecting  the 
extremities  of  the  perpendiculars,  become  a  curve,  called 


INSTRUMENTS  AND   THE    SLIDE  RULE. 


97 


the  logarithmic  curve,  and  from  this  might  every  number 
of  the  slide-rule  be  readily  obtained.  For  practical  pur- 
poses, however,  we  shall  not  require  distances  less  than 
the  eighth  of  an  inch  :  but  it  will  be  advisable  to  deter- 
mine the  lengths  of  the  perpendiculars  by  a  more  tedious 
process  than  the  one  described ;  as  in  this,  the  least  error 
in  drawing  the  parallels  is  so  increased  by  the  divergence 
of  the  lines  forming  the  angle,  as  most  frequently  to  ren- 
der the  curve  altogether  useless. 

From  the  nature  of  the  lines  AD,  AF,  AH,  it  follows 
that  AF  is  a  mean  proportional  between  AD  and  AH. 
Hence,  if  the  lengths  AD  and  AH  were  given,  AF  might 
be  inserted,  by  problem  14,  on  the  parallel  ruler.  Thus, 
to  raise  a  mean  proportional  between  a  b  and  e  f, — 

Produce  e  f,  and  make  f  g  =  a  b ;  bisect  e  g  in  h,  and 
from  the  point  h,  with  the  distance  h  g,  cut  off  f  k.  Bisect 
b  f  in  m,  and  make  m  n  =  f  k.  m  n  is  the  mean  pro- 
portional required. 


In  the  same  way  might  a  mean  proportional  be  placed 


98  A   TREATISE    ON   A   BOX    OF 

halfway  between  m  and  f,  and  again  between  f  and  this, 
and  so  on  to  any  degree  of  exactness. 

Instead  of  bisecting  the  line  b  f  continually,  it  will  le 
better  to  set  up  a  number  of  perpendiculars  first ;  and,  for 
this  purpose,  it  will  be  necessary  to  choose  some  number 
represented  by  2"  -f  1,  as  24  +  1,  2s  +  1  =  17,  33,  &c., 
as  by  this  means  there  will  always  be  a  line  ready,  from 
which  to  cut  off  the  mean  proportional.  Further,  in  ordei 
to  obtain  ten  numbers,  the  last  perpendicular  must  be 
taken  ten  times  the  smaller.  Having,  then,  set  off  any 
distance  16  times  along  AB,  (see  title-page,  Fig.  II.,)  and 
raised  17  perpendiculars,  take  any  height,  AC,  and  com- 
plete the  parallelogram,  ACDB.  Divide  DB  into  ten 
equal  parts  in  the  points  I,  n,  in,  IV,  &c.  Between 
B  1  and  AC,  find  the  mean  proportional  f ;  between  this 
and  AC,  g';  and  so  on,  till  all  are  determined.  Connect 
their  extremities  by  the  logarithmic  curve  C  m  a  1,  and 
through  the  points  IX,  vm,  VII,  VI,  &c.  draw  lines  paral- 
lel to  CD,  to  meet  it  in  m,  k,  h,  e,  &c.;  from  which 
points  draw  the  lines  m  9,  k  8,  &c.  parallel  to  AC.  The 
distances  from  D,  toward  C,  will  represent  the  logarithms 
of  the  numbers  2,  3,  4,  &c.  Lay  this  line  over  again  to 
E,  and  make  EF  equal  to  ED ;  join  FC,  and  draw  the 
parallels;  so  shall  EF  be  divided  logarithmically;  and, 
since  EF  is  twice  CD,  the  numbers  on  CD  shall  be  the 
squares  of  those  on  EF;  and  if  CGr  be  made  equal  to  one- 
third  of  EF,  then  shall  the  numbers  on  CGr  be  the  cubus 
of  those  on  EF,  and  the  square  roots  of  the  cubes  of 
those  on  CD. — Thus,  to  find  the  square  and  cube  of  5. 
Take,  with  the  compasses,  the  distance  from  F  to  5;  this 
will  reach  from  D  to  25,  the  square,  and  from  Gr  to  125, 
the  cube. — To  find  the  square  root  of  16.  Take  16  from 


INSTRUMENTS   AND   THE   SLIDE-RULE.  9U 

D,  it  will  reach  from  F  to  4. — To  find  the  square  root  of 

4  embed.     Measure  from  D  to  4;  the  distance  will  reach 
from  G  to  8. 

The  subdivision  of  the  portions  into  tenths  is  easy ; 
thus,  for  instance,  on  the  line  E  F :  measure  the  distance 
from  1  to  2 ;  this  will  reach  back  from  3  to  1  $ ;  also  from 

5  to  2J,  from  7  to  3£,  and  so  on  :  also  the  distance  from 
1  to  3,  on  DC,  will  reach  forward  from  4  to  12,  from  5 
to  15,  from  6  to  18,  &c. ;  the  distance  from  1  to  4  will 
reach  forward  from  4  to  16,  from  6  to  24,  from  7  to  28, 
&.c. ;  the  distance  from  F  to  2  will  reach  forward  from  G 
to  8,  from  8  to  64,  etc. ;  and  thus,  by  a  little  contrivance, 
may  the  whole  of  the  subdivisions  be  filled  up. 

By  means  of  the  logarithmic  curve  we  may  double  a 
cube  or  globe.  Thus,  suppose  the  diameter  of  a  globe,  or 
the  side  of  a  cube  of  gold,  is  half  an  inch ;  it  is  required 
to  find  the  diameter  of  another  that  shall  contain  twice  as 
much. 


Draw  AB  at  right  angles  to  AC,  and  make  AD  equal 
to  half  an  inch,  and  AE  the  double  of  it.  Draw  EF  and 
DO  parallel  to  AC,  meeting  the  curve  in  F  and  G,  from 


100  A   TREATISE    ON    A   BOX    OF 

which  points  let  fall  the  perpendiculars  FH,  GK.  Di- 
vide HK  into  3  equal  parts,  and  make  NL  and  OM 
parallel  to  FH.  0  M  is  the  diameter  of  the  globe  re- 
quired. 

For,  from  the  nature  of  the  curve, 

KG  :  OM  :  :  OM  :  NL  :  :  NL  :  FH; 
.-.   KG3  :  OM*  :  :  KG  :  FH;  that  is,  as  1  :  2. 

The  duplication  of  the  cube  is  a  problem  famous  in 
antiquity.  It  was  proposed,  by  the  oracle  at  Delphi,  as 
a  means  of  stopping  the  plague  which  was  then  raging  at 
Athens. 

To  lay  the  numbers  down  on  the  rule,  however,  cor- 
rectly, we  must  have  recourse  to  a  table  of  logarithms,  as 
was  shown  in  describing  the  Sector.  The  line  intended 
to  be  numbered  is  to  be  divided  into  1000  equal  parts; 
tfeeri  the  distance  from  1  to  2  will  be  301  of  those  parts, 
this  number  being  the  logarithm  of  2 ;  the  distance  trotn 
1  to  3  will  be  477,  the  logarithm  of  3,  &c. 


INSTRUMENTS   AND   THE   SLIDE-RULE.  101 


THE   SLIDE-RULE. 

THE  Slide-rule  is  an  instrument  containing  the  loga- 
rithmic lines  we  have  been  describing;  they  are  arranged 
in  different  ways,  according  to  the  purpose  for  which  they 
are  intended;  but  the  most  extensively  useful  is  that  in 
which  the  D  line  commences  with  unify.  The  line  A  is 
laid  down  twice  along  the  top  of  the  rule;  the  line  D 
once  in  the  same  space,  at  the  bottom  of  the  rule ;  between 
them  is  a  groove  for  the  reception  of  the  slide  BC,  which 
is  merely  a  copy  of  the  A  line.  In  the  rules  I  have  con- 
structed there  are  two  other  grooves,  for  containing  two 
extra  slides,  when  not  in  use.  One  of  these  is  marked  E, 
and  contains  the  logarithmic  line  repeated  thrice.  The 
third  is  a  trigonometrical  slide,  and  is  graduated  with  lo- 
garithmic sines  and  tangents,  the  former  of  which  work  to 
the  line  D,  and  the  latter  to  A.  At  the  back  is  a  com- 
prehensive table  of  numbers,  suited  to  the  variety  of  lines, 
surfaces,  and  solids  usually  met  with.  In  making  use  of 
the  rule,  it  is  to  be  observed,  that  the  values  of  the  num- 
bers in  the  second  division  of  A,  B,  and  C,  are  ten  times 
those  of  the  first.  If  the  first  series  be  reckoned  as  .01, 
.0-2,  .03,  &c.,  the  second  will  be  .1,  .2,  .3,  &c.  The  first 
.1,  .2,  .3,  &c.,  the  second  1,  2,  3,  &c.  The  first  1,  2,  3, 
&c.,  the  second  10,  20,  30,  &c.  And  here  it  is  immaterial 
whether  a  number  is  chosen  from  the  first  or  second  divi- 
sion ;  but,  in  ascertaining  the  squares  and  square  roots  of 

9* 


102  A    TREATISE    ON   A    BOX    OF 

numbers  with  C  and  D,  it  will  be  necessary  to  observe, 
that  if  the  number  of  figures  representing  the  square  be 
odd,  or  a  decimal  having  an  odd  number  of  ciphers  before 
the  first  effective  figure,  it  must  be  selected  from  the  first 
division  of  C  j  if  otherwise,  from  the  second  division. 
This  is  analogous  to  the  caution  requisite  in  extracting  the 
roots  of  numbers  by  computation,  where  it  is  necessary  to 
make  the  first  point  fall  upon  the  unit,  and  to  have  an 
even  number  of  figures  in  the  decimals.  A  little  practice 
on  the  rule  will  soon  render  this  familiar  :  thus,  to  find 
the  square  soot  of  5,  or  .05,  look  under  the  5  in  the  first 
division  of  C ;  for  the  square  root  of  50,  .5,  or  .005,  under 
5  in  the  second  division  of  C ;  for  in  common  arithmetic 
it  is  necessary  to  put  .5  into  the  shape  of  .50,  and  .005  to 
.0050,  before  we  take  the  root;  and  the  same  form,  of 
course,  applies  to  the  slide-rule.  A  like  principle  applies 
to  the  E  slide.  Whole  numbers  containing  one  figure  are 
in  the  first  division,  two  in  the  second,  and  three  in  the 
third ;  and  decimals  are  managed  accordingly.  Besides 
this,  there  is  a  mutual  relation  between  the  lines,  which 
will  be  readily  understood  by  attending  to  the  remarks 
that  follow. 

When  the  slide  BC  is  laid  evenly  in  the  groove,  that  is, 
when  the  commencement  of  A  coincides  with  the  com- 
mencement of  B,  the  numbers  on  A  are  the  same  as  the 
numbers  on  B;  when  the  slide  is  in  any  other  position, 
the  numbers  on  A  are  proportional  to  the  numbers  on  B. 
The  same  is  the  case  with  the  D  line.  When  the  slide 
lies  evenly  in,  the  numbers  on  C  are  the  squares  of  those 
on  D;  when  the  slide  is  in  any  other  position,  the  numbers 
on  C  are  proportional  to  the  squares  of  those  on  D.  .Thus?, 
lot  the  slide  be  drawn  back  till  the  2  of  B  falls  uuder  the 


INSTRUMENTS   AND   THE    SLIDE-RULE.  103 

1  of  A,  then  we  have  a  series  of  continued  proportionals, 
or  equivalent  fractions,  the  odd  terms  or  numerators  sland- 

1      2      3 

ing  above  the  even  terms  or  denominators,  as  -r  =  -  =- 

u      4      o 

=  t  =  :£r=&c.,orl:2::2:4::3:6::4:8::5:10:: 

o       1U 

,       .1     n      j  TM-       2      8       18      32 

&c. ;  and  on  the  C  and  D  lines  —  =  —  =  —  =  —  =  &c., 

1*       2r       o*        4* 

or  2  :  1s : :  8  :  29 : :  18  :  3a : :  32  :  49 : :  &c.  An  attention 
to  the  above  will  explain  every  operation.  Whenever  we 
require  to  square  a  number,  we  select  this  number  on  D, 
and  look  over  it  on  C ;  whenever  we  wish  to  obtain  the 
square  root  of  a  number,  we  select  the  number  on  C,  and 
look  under  it  on  D.  When  we  neither  require  to  square 
it,  nor  to  extract  the  root,  the  D  line  does  not  enter  into 
the  operation,  and  the  A  and  B  lines  are  used  indiffer- 
ently. We  may  now  proceed  to  the  mode  of  valuing  the 
numbers. 

Let  it  be  required  to  multiply  3  by  5.  Under  the  3 
of  A  set  the  beginning  1  of  the  BC  slide ;  then  over  the 
5  is  15.  Here  the  number  chosen  from  the  first  division 
consisting  of  a  unit,  the  beginning  of  B  also  a  unit,  and 
the  result  falling  in  the  second  division,  it  will  consist 
of  tens. 

To  multiply  3  by  50.  Let  the  slide  stand  as  before; 
then,  the  5  being  taken  as  50,  the  beginning  of  the  slide 
will  be  10 ;  therefore  the  3  standing  over  it,  in  the  first 
division  of  A,  becomes  30;  consequently  the  result,  fall- 
ing in  the  second  division,  will  be  150. 

To  multiply  30  by  50.  Let  the  slide  remain  ;  the  pro- 
duct is  now  1500;  for  the  5  on  B  being  reckoned  f)U,  the 


104  A   TREATISE   ON    A   BOX   OF 

commencement  of  the  slide  will  be  10 ;  therefore  the  30 
standing  over  it  in  the  first  division  becomes  300  :  conse- 
quently the  result  falling  in  the  second  division,  the  pri- 
mary number  will  be  thousands. 

In  dividing  numbers,  the  following  considerations  are 
to  be  attended  to.  If  the  divisor  contain  as  many  figures 
as  the  dividend,  the  beginning  of  the  line  containing  the 
dividend  will  be  unity.  If  the  divisor  have  one  more 
figure,  the  beginning  of  the  line  from  which  the  dividend 
is  chosen  will  be  in  the  Jirst  place  of  decimals;  if  it  have 
two  more,  in  the  second  place  of  decimals  j  if  three  more, 
in  the  third  place  of  decimals ;  and  so  on.  Thus,  divide 

4 
4  by  8,  that  is-  =  ?   Under  4  of  A  in  the  second  division 

o 

place  8  of  B,  denominator  under  numerator  as  in  vulgar 
fractions,  then  over  the  beginning  of  B  is  .5.  For  the 
divisor  containing  only  as  many  figures  as  the  dividend, 
the  beginning  of  the  second  division  of  A  will  be  1,  and 
the  quotient  falling  in  the  first  division,  it  will  be  .5. 

4 

Divide  4  by  80,  that  is  57^=  ?     Let  the  slide  remain ; 
oO 

here,  the  divisor  containing  one  more  figure  than  the  divi- 
dend, the  beginning  of  the  second  division  will  be  in  the 
first  place  of  decimals,  that  is  .1,  consequently  the  quo- 
tient will  be  .05. 

4 

Divide  4  by  800,  that  is  U7-^  =  ?    Let  the  slide  stand ; 
oOO 

here,  the  divisor  containing  two  more  figures  than  the 
dividend,  the  beginning  of  the  second  division  will  be  in 
the  second  place  of  decimals,  that  is  .01,  consequently  the 
quotient  will  be  .005. 


INSTRUMENTS   AND    THE    SLIDE  RULE.  105 

The  numbers  on  C  being  the  squares  of  those  on  D,  if 
the  beginning  of  D  is  10,  that  of  C  will  be  100. 

Q  \x  ga 
Divide  by  40,  3  times  8  squared,  that  is     *     =?  To 

40  on  A  set  3  of  B,  over  8  of  D  is  4.8.  Here,  40  con- 
taining one  more  figure  than  the  3,  the  3  becomes  .3, 
consequently  the  result,  falling  in  the  next  division,  will 
be  4.8. 

3  X  80s 
Divide  by  40,  3  times  80  squared,  that  is  — -^ —  =  ? 

Let  the  slide  remain;  here  the  3  becomes  .3  as  before, 
but  the  commencement  of  D  being  reckoned  as  10,  the 
D  umbers  on  C  are  increased  100  times,  and  therefore  the 
.3  becomes  100  times  .3  or  30 ;  hence  the  result,  falling 
in  the  next  division,  will  be  480. 

To  divide  by  decimals,  add  as  many  ciphers  to  the  di- 
vidend as  the  first  effective  figure  is  removed  from  the 

decimal  point;  thus  -$  =  1   To  10  of  A  place  .2  of  B;  the 

2  is  in  tbejirst  place  of  decimals,  therefore  add  one  cipher 
to  the  10  and  it  becomes  100.  Look  back  to  the  begin- 
ning of  the  slide,  and  over  it  is  50. 

1  fi 
What  is  the  value  of  —  ?   To  16  of  A  set  4  of  B  j  the 

4  being  the  second  place  of  decimals,  add  two  ciphers,  and 
we  have  1600  for  the  value  of  the  16.  Look  back  to  the 
beginning  of  the  slide,  over  which  is  400. 

Q  \/  .102 

Find  the  value  of     *    -  .    To  8  on  A  set  3  of  B,  over 
.US 

40  of  D  is  60,000.  Here  the  8  of  .08  being  the  second 
place  of  decimals,  add  tiro  ciphers,  smd  the  3  becomes  300, 


106  A   TREATISE   ON   A   BOX   OF 

(3  with  two  ciphers.)  Again,  the  beginning  of  D  being 
10,  the  numbers  on  C  become  increased  by  102,  or  100,  so 
that  two  more  ciphers  have  to  be  added,  and  the  3  becomes 
3  with  four  ciphers,  that  is  30,000  j  consequently  the  re- 
sult falling  more  to  the  right  will  be  60,000. 

7  V  40a 
Find  the  value  of  — ~ — ,  that  is,  divide  by  60a, 

7  times  40a.  To  60  of  D  set  7  of  C,  over  40  is  3.111. 
Here  60  and  40,  the  two  numbers  selected  on  D,  having 
each  the  same  number  of  figures,  the  result  falling  on  C 
will  contain  the  same  number  of  integers  as  the  7  ;  hence 
the  quotient  is  3.111. 

Divide  by  602,  7  times  42,  that  is     *    ' '  =  ?  Let  the 

slide  stand  as  before.  Here  the  4  of  the  D  line  contain- 
ing one  figure  less  than  the  60,  the  square  of  it  will  con- 
tain two  figures  less,  consequently  the  7  over  the  60  is 
reduced  to  the  second  place  of  decimals,  and  becomes  .07; 
hence  the  result  will  be  .03111. 

Divide  by  203,  70  times  50.  Over  20  on  D  set  70  of 
the  slide,  and  look  under  50  of  A.  Here,  the  commence- 
ment of  the  D  line  being  10,  the  beginning  of  the  A  line 
would  be  100 ;  but  if  we  select  50  from  the  first  division 
of  A,  we  reduce  it  one-tenth,  and  the  70  becomes  7 ;  con- 
sequently the  number  under  50  of  A  is  8.75. 

These  exercises,  if  carefully  attended  to,  will  be  amply 
sufficient  to  enable  the  student  to  value  all  quantities 
correctly. 

A  very  common  operation  on  the  slide-rule  is  to  find  a 
mean  proportional  between  two  numbers,  that  is,  to  ex- 


INSTRUMENTS   AND   THE   SLIDE-RULE.  107 

tract  the  square  root  of  their  product.  Let  the  two  num- 
bers be  4  and  9.  Multiply  4  by  Q,  and  take  the  root; 
that  is,  to  4  of  A  set  commencement  of  slide,  under  9  is 
6.  As  the  slide  stands,  it  necessarily  follows  that  4  of 
the  slide  is  over  4  of  D,  since  4  is  the  square  root  of  4 
times  4 ;  hence,  in  finding  the  mean  proportional  between 
two  numbers,  it  may  be  effected  with  the  C  and  D  lines 
only,  by  setting  one  of  the  numbers  over  itself,  and  look- 
ing under  the  other.  Thus,  what  is  the  mean  proportional 
between  3  and  12  ?  Set  3  over  3,  then  under  12  is  6. 
The  object  intended  to  be  accomplished  by  finding  this 
mean  proportional,  is  to  reduce  parallelograms  to  squares, 
and  ellipses  to  circles  j  a  square  whose  side  is  6  inches 
being  equal  to  a  parallelogram  12  inches  by  3,  and  an 
ellipse,  whose  axes  are  12  and  3,  equal  to  a  circle  whose 
diameter  is  6. 

As  we  shall  have  occasion  hereafter  to  introduce  various 
formulae  for  solids,  it  will  be  necessary  for  the  learner  to 
study  the  following  operations : — 

g     /Q8         I          rj»"\ 

Find  the  value  of  — - — ^ — - .      Now,  this  is  equal  to 

g    vx   09  g   vx   KS 

— 1-  -       — .       Therefore  it  must  be  effected  by 

obtaining  the  quotients  separately,  and  then  adding  them 
together.  Hence,  over  the  6  of  D  set  8  of  the  slide, 
then — 

Over  3  is  2. 

Over  5  is  5.55 

L55 


108  A   TREATISE   ON    A   BOX   OF 


40  f>-l9  _U  v  v  398^ 

Find  the  value  of  -  -;.    Over  33  of  D 

oo2 

set  40,  then  — 

Over  24  is  21.2 

Over  32  is  37.6 

Ditto       37.6 


96.4 

,  40  (24a  +  323  +  59.22) 

Find  the  value  of  — 4£f       -.   Over 46  of 

4o9 

D  set  40,  then — 

Over  24  is  10.9 

32  is  19.35 
59.2  is  66.2 


96.45 

It  sometimes  happens  that  we  require  to  multiply  three 
numbers  together.  This  cannot  be  done  by  the  kind  of 
rule  we  have  been  considering  in  one  operation,  but  it 
may  be  effected  by  dividing  by  the  reciprocal  of  one  of  the 
numbers.  Thus,  let  it  be  required  to  find  the  product  of 

4X7X8.     The  reciprocal  of  4  is  T  =  25  j  hence  we 

have  to  divide  by  .25,  7  times  8.  To  .25  on  A  set  7  of 
the  slide,  under  8  on  A  is  224.  By  inverting  the  slide, 
and  pushing  it  evenly  in,  that  is,  until  the  end  is  under 
the  beginning  of  A,  it  will  be  seen  that  the  numbers  on  B 
are  the  reciprocals  of  the  corresponding  ones  on  A ;  hence 
if  instead  of  the  D  line,  a  line  similar  to  A  were  laid  down 
under  the  slide,  in  an  inverted  position,  it  would  furnish  a 
series  of  reciprocals,  and  then  three  numbers  might  at  once 


. 

IXSTKUMENTS    AND    THE    SLIDE-RULE.  lO'A 

be  multiplied  together,  by  taking  one  of  them  on  this  in- 
verted line,  one  on  the  slide,  and  the  third  on  the  A  line, 
under  which  would  be  the  product.  Moreover,  if,  instead 
of  laying  it  down  so  as  to  make  the  commencement  of  it 
fall  under  the  end  of  the  slide,  it  were  drawn  out  toward 
the  right  hand  till  some  other  number  than  unity  stood 
under  the  end  of  the  A  line,  then  the  product  of  the  three 
numbers  would  be  divided  by  this  constant  number.  For 
instance,  supposed  we  wished  to  divide  by  2,  7  times 

IT  \s  8^6 

8  times  6  ;  that  is,  to  find  the  value  of  -          — .    Let 

a 

the  inverted  line  be  placed  so  that  the  2  shall  fall  under 
the  end  of  the  A  line ;  then  over  the  7  of  this  inverted 
line  place  8  of  the  slide,  and  under  the  6  of  A  will  be  168. 
Hence,  if  a  person  pursued  an  occupation  in  which  his 
calculations  required  to  be  divided  by  a  constant  number, 
he  might  have  a  rule  constructed  to  suit  himself  for  that 
particular  number.  A  few  such  rules  are  in  use.  The 
officers  of  the  customs  have  frequently  to  measure  pieces 
of  timber,  the  length  of  which  is  taken  in  feet,  and  the 
breadth  and  thickness  in  inches.  Now,  multiplying  these 
three  dimensions  together,  and  dividing  by  144,  gives  the 
solidity  in  cubic  feet.  Hence  let  the  A,  B,  and  C  lines  be 
laid  down  as  Us'ial,  and  instead  of  D  substitute  an  inverted 
A  line,  so  placed  that  144  shall  fall  under  the  end  of  the 
slide.  Then,  if  a  piece  of  timber  measures  55  feet  long, 
24  inches  broad,  and  9  thick ;  under  55  of  A  place  24  of 
the  slide,  and  over  9  of  the  inverted  line  is  82^  cubic  feet, 
the  content.  In  malt  guaging  again,  the  number  of  cubic 
indies  in  a  bushel  is  2218.19.  Hence,  taking  the  di- 
mensions in  inches,  let  the  inverted  line  be  so  placed  that 
the  nuiii'.c;-  ^>^1>.  1'J  shall  fall  under  the  end  of  the  slide; 


110  A   TREATISE   ON   A  BOX  OP 

then  if  a  cistern  of  malt  measures  30  inches  long,  16 
broad,  and  12  deep ;  to  30  on  A  set  16  of  the  slide,  and 
over  12  of  the  inverted  line  is  2.6  nearly,  the  content  in 
bushels.  If  we  wish  to  obtain  the  result  in  gallons,  (as 
8  gallons  make  a  bushel,)  take  8  times  one  of  the  dimen- 
sions :  for  instance,  to  240  on  A  set  16  of  the  slide,  and 
over  12  of  the  inverted  line  is  20.78  gallons. 

In  practice  these  rules  are  of  the  utmost  convenience 
possible,  and  the  principle  might  be  carried  out  with  ad- 
vantage to  a  much  greater  extent  than  it  yet  has  been. 


OBSERVATIONS. 

There  are  three  kinds  of  measure — lineal,  superficial, 
and  solid :  lineal,  for  such  things  as  have  length  only ; 
superficial,  for  those  that  have  length  and  breadth ;  and 
solid,  where  there  are  length,  breadth,  and  thickness. 
When  lines  vary  proportionally  they  vary  simply  as  then 
measures ;  when  surfaces  vary  proportionally  they  vary  as 
the  squares  of  their  like  measures ;  and  when  solids  varj 
proportionally  they  vary  as  the  cubes  of  their  like  measures. 
Thus,  let  there  be  two  similar  funnels,  or  cones,  A  and  B ; 
and  let  A  be  filled  with  water  to  the  depth  of  1  foot,  and 
B  to  the  depth  of  2  feet ;  then  the  circumference  of  the 
top  of  the  water  in  B  will  be  twice  that  of  A,  both  being 
lines;  the  area  of  the  top  of  the  water  in  B  will  be  4 
times  that  of  A,  or  2a,  both  being  surfaces:  and  the 
weight  or  quantity  of  the  water  in  B  will  be  8  times  that 
of  A,  or  23,  both  being  solids :  and  so  of  all  surfaces  and 
polids  that  vary  proportionally. 

If  a  number  of  regular  polygons  have  oqual  periinetero. 


INSTRUMENTS   AND   THE   SLIDE-RULE.  Ill 

tfiut  contains  the  greatest  amount  of  surface  in  which  the 
perimeter  is  distributed  among  the  greatest  number  of 
sides ;  and,  as  a  circle  may  be  conceived  to  be  a  polygon 
of  an  infinite  number  of  sides,  it  therefore  contains  the 
greatest  quantity  of  space  within  the  shortest  bounding 
line. 

A  regular  polygon  contains  more  than  an  irregular 
polygon  of  the  same  number  of  sides,  their  perimeters 
being  equal ;  thus,  an  equilateral  triangle  has  a  greater 
area  than  any  other  triangle  of  equal  ambit ;  and  a  square 
is  the  largest  quadrilateral  that  can  be  constructed  with 
sections  of  the  same  line. 

In  the  same  way  as  the  circle  contains  the  largest  sur- 
face within  the  least  compass,  so  the  sphere  contains  the 
greatest  bulk  within  the  smallest  space. 


RATIOS  AND  GAUGE  POINTS. 

At  the  back  of  the  rule  will  be  found  a  quantity  of 
tabular  work,  adapted  to  various  kinds  of  calculation  : 
these  consist  of  ratios  and  gauge  points.  Ratios  express 
the  proportions  existing  between  certain  lines,  or  num- 
bers ;  thus,  if  the  diameter  of  a  circle  be  113,  its  circum- 
ference will  be  355;  and,  as  the  circumference  varies  as 
the  diameter,  therefore  113  :  355  expresses  the  ratio  of 
the  diameter  of  any  circle  to  its  circumference.  Gauge 
points  are  the  square  roots  of  divisors ;  thus,  if  we  require 
to  reduce  square  inches  to  square  feet,  we  must  divide  by 
144,  which  number  may  be  chosen  on  A;  if  instead  of 
this  we  divide  by  12a,  we  take  12  upon  the  D  line,  and, 


112  A   TREATISE   ON   A   BOX   OP 

for  the  sake  of  distinction,  12  is  called  a  gauge  point  In 
rules  having  the  D  line  commencing  with  unity,  when  the 
slide  is  set  to  any  gauge  point,  it  is  also  set  to  the  cor- 
responding divisor,  the  one  standing  under  the  slide,  the 
other  above  it ;  and  therefore,  with  such  rules,  it  would 
be  immaterial  whether  we  used  divisors  or  gauge  points; 
as  however,  the  formulae  for  many  surfaces,  and  almost  all 
solids,  require  the  use  of  the  D  line,  it  is  far  more  conve- 
nient for  valuing  the  numbers,  to  make  use  of  gauge- 
points,  and  therefore  the  tabular  work  is  so  constructed. 


TABLE  I.  contains  a  list  of  ratios  belonging  to  the 
circle,  commencing — 

A  B 

113  diameter  =  355  circumference. 
44  diameter  =    39  side  of  equal  square. 

That  is,  under  113  on  A  set  355  on  B,  then  the  num- 
bers on  A  will  be  a  series  of  diameters,  and  the  numbe,  3 
beneath  them  on  B  will  be  their  corresponding  circumfe- 
rences ;  and  so  of  all  the  rest. 

EXAMPLES. 

1.  If  the  diameter  of  a  circle  is  8  inches,  what  is  its 
circumference  ?     Set  the  rule  as  directed,  then  under  8  is 
25.13  inches. 

2.  The  diameter  of  a  circle  is  9  inches,  what  is  the  side 
of  an  equal  square  ?     Under  44  of  A  set  39  of  B ;  under 
9  is  7.97  inches. 


INSTRUMENTS  AND  THE   SLIDE-RULE.  113 

S.  The  radius  of  a  circle  is  6  inches,  what  is  the  leijgth 
of  an  arc  of  it  containing  31 J  degrees?  Under  57.3  de- 
grees on  A  set  6  inches  on  B ;  under  31  £  degrees  on  A 
is  3.3  inches. 

4.  The  circumference  of  a  circle  is  75,  what  is  the  di- 
ameter ?— Ans.  23.87. 

5.  The  diameter  is  7,  what  is  the  circumference  ? — Ans. 
22  nearly. 

6.  The  diameter  is  17,  what  is  the  circumference  ? — 
Ans.  53.4. 

7.  Suppose  the  diameter  of  the  earth  to  be  7960  miles, 
what  is  its  circumference  ? — Ans.  25,000  miles. 

8.  The  diameter  of  a  circle  is  6  inches,  what  is  the  side 
of  a  square  inscribed  within  it  ? — Ans.  4.24. 

9.  The  circumference  of  a  circle  is  12  feet,  what  is  the 
side  of  an  equal  square  ? — Ans.  3.38. 

10.  The  circumference  of  a  circle  is  15  inches,  what  is 
the  side  of  its  inscribed  square  ? — Ans.  3.375. 

11.  The  side  of  a  square  is  10  inches,  what  is  the 
diameter  of  an  equal  circle  ? — Ans.  11.28. 

12.  The  side  of  a  square  is  20  yards,  what  is  the  cir- 
cumference of  an  equal  circle  ? — Ans.  70.83. 

13.  The  side  of  a  square  is  19  inches,  what  is  the  side 
of  an  equal  equilateral  triangle  ? — Ans.  28.88. 

14.  The  area  of  a  circle  is  27,  what  is  the  area  of  a 
square  inscribed  in  it? — Ans.  17.18. 

lu* 


114  A  TREATISE   ON   A   BOX    OF 

15.  An  arc  of  38  degrees  measures  5  inches,  irO'  iJ 
the  radius  of  the  circle  of  which  it  is  a  part? — Aa»  "?.(><>. 

16.  I  have  a  circular  piece  of  wood,  whose  diameter  ig 
15  inches,  and  wish  to  cut  the  largest  square  out  of  it ; 
what  will  be  the  length  of  each  side  ? — Ans.  10.6  inches. 

The  method  of  obtaining  these  ratios  in  whole  numbers 
is  a  beautiful  exemplification  of  the  abridgment  of  labour 
effected  by  the  slide-rule ;  and  of  performing,  with  the 
utmost  facility,  operations  that  would  require  considerable 
time  and  trouble  by  any  other  means.  Archimedes  dis- 
covered that  the  ratio  of  7  to  22  nearly  expressed  that  of 
the  diameter  of  a  circle  to  its  circumference.  Purbachius, 
in  the  fifteenth  century,  making  the  diameter  120,  reck- 
oned the  circumference  at  377.  Metius,  two  centuries 
later,  subtracted  the  7  and  22  from  the  120  and  377,  and 
obtained  the  numbers  113  and  355.  This  last  ratio 
is  easily  remembered,  from  its  containing  the  first  three 
odd  numbers  in  pairs,  and  it  is  remarkably  accurate,  the 
quotient  of  355  by  113  being  true  to  the  sixth  place  of 
decimals.  The  obtaining  of  these  ratios  in  integers,  how- 
ever, must  have  been  a  task  of  considerable  labour.  To 
determine  them  by  the  slide-rule  is  the  work  of  a  moment. 
By  various  modes  of  computation  it  may  be  shown  that  if 
the  diameter  be  1,  the  circumference  will  be  nearly 
3.1416 ;  therefore,  under  1  of  A  set  3.1416,  as  nearly  as 
possible,  and  run  the  eye  along  until  you  find  two  num- 
bers coinciding :  such  will  be  113  and  355,  which  will 
be  the  ratio  required.  The  advantage  of  having  the  ratios 
in  whole  numbers,  for  the  purposes  of  the  slide-rule,  is 
obvious,  as  they  can  be  set  with  greater  rapidity  and 
exactness  than  decimals. 


INSTRUMENTS    AND   THE    SLIDE-RULE. 


115 


The  following  table  will  enable  the  student  to  solve  the  pr«- 
questions  numerically : — 

Diameter  1,  circumference  3.1416.' 

side  of  equal  square,  .8862. 

side  of  inscribed  square  .7071. 

Circumference  1,  diameter  .3183. 

side  of  equal  square  .2821. 

side  of  inscribed  square  .2251. 

Side  of  square  1,  diameter  of  equal  circle  1.128. 
circumference  of  equal  circle  3.545. 

side  of  equal  equilateral  triangle  1.5196. 

Area  of  circle  1,  area  of  inscribed  square  .6366. 
Area  of  square  1,  area  of  inscribed  circle  .7854. 

area  of  inscribed  octagon  .8284. 

The  length  of  an  arc  of  57.2957795  degrees  =  radius  of  circle 

Solution  of  question  8:— .7071  X   6  =  4.2426. 


TABLE  II.  contains  the  Linear  Dimensions  of  Polygons 
described  within  and  without  Circles,  and  commences 
thus : — 


No.  of 

Sides. 

Inscribed  Polygon. 

Circumscribed 
Polygon. 

A. 

Diam. 

B. 

Side. 

A. 

Diam. 

B. 

Side. 

3 
4 

15 

9.9 

13 

7 

15 
1 

26 
1 

That  is,  if  the  diameter  of  a  circle  be  15,  the  side  of  an 
equilateral  triangle  inscribed  within  it  will  be  13  :  hence, 
under  15  of  A  set  13  of  B;  then  the  numbers  on  A  will 
be  a  series  of  diameters,  and  the  numbers  beneath  them 
on  B  will  be  the  sides  of  the  corresponding  triangles;  and 
so  of  the  rest.  The  method  of  obtaining  them  is  first 
by  computation,  and  then  as  for  the  ratios  before 
described. 


116 


A   TREATISE   ON  A   BOX   OF 


EXAMPLES. 

17.  The  diameter  of  a  circle  is  12  inches,  what  will  bo 
the  side  of  an  equilateral  triangle  inscribed  therein  ? — 
Under  15  of  A  set  13  of  B;  under  12  of  A  is  10.4 
inches. 

18.  The  diameter  of  a  circle  is  11 J  inches,  what  is  the 
side  of  a  regular  pentagon  inscribed  within  it  ? — Ans.  6.76. 

19.  A  circle  whose  diameter  is  9i  inches  has  a  regular 
hexagon  surrounding  it,  what  is  the  length  of  each  side  ? 
—Ans.  5.33. 

20.  A  person  having  a  circular  piece  of  ground  37 
yards  in  diameter,  wishes  to  make  within  it  a  flower-bed 
of  a  heptagonal  form,  whose  area  shall  be  a  maximum ; 
what  will  be  the  length  of  each  side  ? — Ans.  16. 

21.  If  I  make  the  diameter  of  a  circle  a  parallel  dis- 
tance on  the  line  L  of  the  sector  from  100  to  100,  what 
parallel  distance  must  I  take  off  as  the  side  of  an  undoca- 
gon  inscribable  therein  ? — Ans.  28.1. 

The  following  table  will  enable  the  student  to  solve  the  pre 
ceding  questions  numerically. 

The  diameter  of  the  circle  being  unity, 


No.  of 
Sides. 

Side  of  Inscribed 
Polygon. 

Side  of  Circumscribed 
Polygon. 

3 

.8660254 

1.7320508 

4 

.7071068 

1.0000000 

5 

.5877853 

.7265425 

6 

.5000000 

.5773503 

7 

.4338837 

.4815745 

8 

.3826834 

.4142136 

9 

.3420201 

.3639702 

10 

.3090170 

.3249197 

11 

.2817325 

.2936264 

12 

.2588190 

.2679492 

Solution  of  question  20:— .4338837  X  37  =  16.0536969. 


INSTRUMENTS  AND    THE    SLIDE-RULE. 


117 


TABLE  III.  contains  the  Areas  of  Polygons,  commenc- 
ing thus : — 


No.  of 
Sides. 

C. 

Area. 

D. 

Side. 

3 
5 

3.9 
43 

3 
5 

That  is,  if  3  be  the  side  of  an  equilateral  triangle,  its  area 
will  be  3.9,  and,  as  similar  surfaces  vary  as  the  squares 
of  their  like  measures,  if  over  3  of  D  we  set  3.9  on  C, 
then  the  numbers  on  D  will  be  a  series  of  sides,  and  the 
numbers  over  them  on  C  their  corresponding  areas. 

EXAMPLES. 

22.  The  side  of  an  equilateral  triangle  is  2,  what  is  its 
area  ?     Over  3  of  D  set  3.9  on  C ;  over  2  of  D  is  1.732, 
the  area  required. 

23.  Required  the  area  of  a  regular  nonagon  having  a 
side  of  7.3  yards.— Ans.  329  yds. 

24.  What  is  the  area  of  an  undecagon  whose  side 
measures  6.4  feet  ? — Ans.  383.6  feet. 

25.  The  side  of  an  octagon  is  4.9  feet,  what  is  its  area  ? 
— Ans.  116  nearly. 

The  side  being  given  in  inches,  to  find  the  area  in  square 
feet,  take  12  times  the  number  on  D,  for  the  number  of 
inches  in  a  square  foot  is  122. 

EXAMPLES. 

26.  The  side  of  an  equilateral  triangle  is  19  inches, 
how   many  square  feet  does  it  contain  ? — Over  36  of  D 
set  3.9  of  C  j  over  19  of  D  is  1.086  feet. 


118  A   TREATISE   ON  A   BOX   OF 

27.  The  side  of  a  regular  pentagon  is  53  inches,  how 
many  square  feet  does  it  contain? — Ans.  33.55. 

28.  What  is  the  area  of  a  nonagon,  each  of  whose  sides 
measures  27  inches? — Ans.  31.29. 

The  side  being  given  in  feet,  to  find  the  area  in  square 
yards,  take  3  times  the  number  on  D,  for  the  number  of 
feet  in  a  square  yard  is  3a. 

EXAMPLES. 

29.  The  side  of  a  regular  pentagon  is  7  feet,  how  many 
square  yards  does  it  contain  ? — Over  15  of  D  set  43  of  C; 
over  7  of  D  is  9.36  yards. 

30.  What  is  the  area  of  a  heptagon  whose  side  measures 
17  feet?— Ans.  116.7  yards. 

31.  A  decagon   measures  20.2  feet  along  each  side, 
what  is  the  area? — Ans.  348.8  yds. 

The  following  table  will  enable  the  student  to  solve  the  pre- 
ceding questions  numerically.  Multiply  the  subjoined  numbers 
by  the  square  of  the  side. 

Equilateral  Triangle 4330127 

Pentagon    1.7204774 

Hexagon. 2.5980762 

Heptagon    3.6339124 

Octagon 4.8284272 

Nonagon 6.1818242 

Decagon 7.6942088 

Undecagon 9.3656411 

Dodecagon 11.1961524 

Solution  of  Question  22:— .4330127  X  2s  =  1.7320508. 


INSTRUMENTS   AND   THE    SLIDE-RULE.  119 

FALLING  BODIES. 

TABLE  IV.  is  of  a  miscellaneous  nature,  and  will  be 
understood  by  inspection.  It  is  found  that  a  heavy  body, 
in  the  latitude  of  London,  falls  579  feet,  or  193  yards  in 
6  seconds ;  and  the  spaces  descended  by  falling  are  as  the 
squares  of  the  times ;  hence,  as  directed  by  the  table,  over 
6  seconds  on  D  set  579  feet  on  C,  (or  193  yards,  if  the 
distance  be  required  in  yards,)  then  the  numbers  on  G 
will  be  a  series  of  distances  fallen,  and  the  numbers  be- 
neath them  on  D  the  seconds  elapsed  in  falling.  The 
same  law  applies  to  bodies  projected  directly  upwards,  the 
retardation  corresponding  with  the  acceleration  in  an 
inverse  order. 

EXAMPLES. 

32.  How  many  feet  will  a  body  fall  in  1  second  ?— 
Over  6  of  D  set  579  on  C;  over  1  of  D  is  16^  feet. 

33.  If  a  ball  is  propelled  straight  upwards,  and  is  found 
to  be  18  seconds  before  it  again  falls  to  the  earth,  how 
many  yards  has  it  ranged  ? — 9  seconds  occupied  in  ascend- 
ing, 9  in  descending ;  over  6  of  D  set  193  on  C ;  over  9 
is  434  yards. 

34.  Standing  at  the  mouth  of  a  well,  which,  by  means 
of  reflecting  the  sun's  rays  into  it  with  a  mirror,  I  per- 
ceived to  be  of  considerable  depth,  I  dropped  a  stone  into 
it,  and  found  it  reached  the  water  in  3J  seconds;  what 
was  its  depth  ? — Ans.  197  feet. 

35.  How  long  would  a  cannon-ball,  fired  perpendicu- 
larly upwards,  be  in  rising  a  mile,  if  it  went  no  higher  ? — 
Ans.  18.12  seconds. 


A   TREATISE   ON    A    BOX   01 


PENDULUMS. 

A  pendulum  22  inches  long,  as  shown  by  the  table, 
makes  80  vibrations,  or  40  revolutions  per  minute,  and 
their  lengths  vary  reciprocally  as  the  squares  of  their 
times,  their  velocity  being  regulated  by  the  force  of 
gravity,  like  that  of  falling  bodies.  Hence,  invert  the 
slide,  and  set  22  inches  on  B  over  80  vibrations,  or  40 
revolutions  on  D  ;  then  the  numbers  on  the  inverted  line 
will  be  a  series  of  lengths,  and  the  numbers  beneath  them 
on  D  the  corresponding  number  of  vibrations  or  revolu- 
tions. 

EXAMPLES. 

36.  What  is  the  length  of  a  pendulum  vibrating  60 
times  per  minute  ?  —  Over  60  is  39.2  inches. 

37.  What  is  the  length  of  a  pendulum  vibrating  64 
times  per  minute  ?  —  Ans.  34.4  inches. 

38.  What  is  the  length  of  a  pendulum  making  29  re- 
volutions per  minute  ?  —  Ans.  42  inches. 

EXPERIMENT. 

Suspend  from  a  hook  in  the  ceiling  a  string  with  a 
bullet  at  the  endj  set  it  vibrating,  or  swing  it  round  so 
as  to  cause  it  to  revolve,  and  compare  its  motions  with  a 
watch. 


The  next  part  of  the  tabular  work  relates  to  the  areas 
rf  circles  and  surfaces  of  spheres,  and  is  as  follows : — 
Circle  43  area  C  =  7.4  diameter  D. 

23  area  C  ^  17  circumference  D. 
Sphere  172  surface  C  =  7.4  diameter  D. 
92  surface  C  =  17  circumference. 


INSTRUMENTS  AND   THE   SLIDE-RULE.  121 

The  student  will  perceive  from  this  that  the  surface  of 
a  sphere  is  4  times  the  area  of  a  circle  of  equal  diameter 
That  is,  if  an  orange  were  perfectly  round,  and  cut  into 
two  equal  parts,  then  the  external  surface  of  the  rind  in 
each  half  would  be  just  double  the  surface  of  the  part  cut 
by  the  knife.  Similar  surfaces  varying  as  the  squares  of 
their  like  measures,  the  dimensions  being  taken  on  D  the 
areas  will  be  on  C. 

EXAMPLES. 

39.  The  diameter  of  a  circle  is  5  inches,  what  is  its 
aiea? — Over  7.4  of  D  set  43  of  C;  over  5  is  19.63  square 
inches. 

40.  What  is  the  area  of  a  circle  whose  circumference  is 
12  inches? — Ans.  11.46  inches. 

41.  The  circumference  of  a  sphere  is  12  inches,  what 
is  the  surface  ? — Ans.  45.8  square  inches. 

If  the  dimensions  are  given  in  inches  and  the  area  is 
required  in  feet,  take  12  times  the  number  on  D;  and  if 
the  dimensions  are  in  feet,  and  the  area  is  required  in 
square  yards,  take  3  times  the  number  on  D. 

EXAMPLES. 

42.  The  diameter  of  a  circle  is  19  inches,  what  is  its 
area  in  square  feet?     12  times  7.4  =  88.8;  over  88.8  on 
D  set  43  on  C ;  over  19  is  1.97  square  feet. 

43.  The  circumference  of  a  circle  is  43  inches,  how 
many  square  feet  does  it  contain? — Ans.  1.021  square 
feet. 

44.  The  diameter  of  a  sphere  is  17  feet;  what  is  its 
surface  in  square  yards? — Ans.  100.8  square  yards. 


122  A  TREATISE   ON   A  BOX   OP 

The  nezt  part  of  the  table  is — 

1  C  =  side  of  square  or  cube  D. 

2  G  =  diagonal  of  square  or  diameter  of  circumscribing 

circle  D. 

8  C  =  diagonal  of  cube  or  diameter  of  circumscribing 
sphere  D. 

That  is,  set  1  of  C  over  the  side  of  a  given  square  on  "D, 
then  under  2  of  C  will  be  its  diagonal,  or  the  diameter  of 
its  circumscribing  circle. 

EXAMPLES. 

45.  A  circle  12  inches  in  diameter  has  a  square  in- 
scribed within  it,  what  is  the  length  of  each  side  ? — Over 
12  of  D  set  2  of  C;  under  1  of  C  is  8.48  inches. 

46.  A  cube  measures  7  inches  along  the  side,  what  will 
be  the  diagonal  of  the  face,  and  what  of  the  cube? — 

Ans.  9.9  diagonal  of  the  face. 
12.12  diagonal  of  the  cube. 

47.  What  is  the  longest  line  that  can  be  taken  in  a 
cubical  box  whose  sides  measure  19.4  feet? — Ans.  33.6 
feet. 

48.  A  square  inscribed  in  a  circle  measures  43  inches, 
what  is  the  diameter  of  the  circle? — Ans.  60.8  inches. 

49.  The  diameter  of  a  sphere  is  26.7  inches;  what  will 
be  the  side  of  the  largest  cube  that  can  be  cut  from  it  ? — 
Ans.  15.41. 

50.  Standing  within  a  cubical  room  I  found  that  the 
distance  from  one  of  the  top  corners  to  the  opposite  cor- 
ner at  bottom  was  23.3  feet;  what  was  the  distance  of 
the  ceiling  from  the  floor? — Ans.  13.45  feet. 


INSTRUMENTS   AND   THE   SLIDE-RULE.  123 


VELOCITY  OF  SOUND. 

THE  flight  of  sound  is  uniformly  proportional  to  the 
time;  hence  use  the  A  and  B  lines  as  directed  in  the 
table. 

EXAMPLES. 

51.  I  observed  the  flash  of  a  gun  12  seconds  before 
hearing  the  report;  how  far  was  it  distant  from  me? — 
To  65  seconds  on  A  set  14  miles  on  B ;  under  12  of  A  is 
2.59  miles  on  B. 

52.  I  observed  a  flash  of  lightning,  and  7  seconds  after- 
wards heard  the  thunder ;  how  far  distant  was  the  electric 
cloud? — Ans.  Ij  mile. 

53.  A  person  standing  on  the  bank  of  a  river,  heard 
the  echo  of  his  voice  reflected  from  a  rock  on  the  opposite 
bank  in  5  seconds  after;  what  was  the  breadth  of  the 
river? — Ans.  950  yards. 

The  subjoined  table  will  enable  the  student  to  solve  the  ques- 
tions by  computation : — 

A  body  falls  162'T  feet  in  the  first  second. 

A  pendulum  vibrating  seconds  in  the  latitude  of  London,  is 
89.1*396  inches. 

In  a  pendulum  describing  a  conical  surface,  the  time  of  revo- 
lution is  equal  to  the  time  of  two  oscillations  of  a  simple  pen- 
dulum, equal  to  the  height  of  the  cone:  that  is,  a  pendulum 
takes  the  same  time  in  going  half  round  a  circle  as  it  does  in 
falling  across  it. 

Putting  d  diameter,  c  circumference, 

Area  of  Circle  =  .7854  d*  or  .07958  c». 
Surface  of  Sphere  =  3.1416  d«  or  .31832  c'. 


124  A  TREATISE   ON   A   BOX   OP 

Putting  s  side  of  square  or  cube,  then  diagonal  of  square,  or 
diameter  of  circumscribing  circle  =  av/2  =  «X  1.4142136. 

Diagonal  of  cube,  or  diameter  of  circumscribing 

sphere  =  s  ^/  3  =  s  X  1.7320508. 
Sound  flies  about  380  yards  per  second. 
Solution  of  Question  33  .16T'?  X  9a  =  1302f  feet  =  434j  yards. 


SURFACES. 

WE  now  come  to  Table  5,  which  consists  of  a  number 
of  gauge  points  for  the  mensuration  of  surfaces,  quadri- 
laterals, triangles,  parabolas,  circles,  cycloids,  and  ellipses ; 
and  the  surfaces  of  prisms,  cylinders,  pyramids,  cones,  and 
spheres.  The  area  of  a  rectangle  is  equal  to  the  product 
of  the  length  and  breadth.  The  area  of  a  trapezoid  is 
found  by  multiplying  half  the  sum  of  the  parallel  sides  by 
the  perpendicular  distance  between  them.  A  triangle  is 
half  a  rectangle,  and  therefore  its  area  is  half  the  product 
of  the  height  and  base.*  The  areas  of  trapeziums  and 
multilaterals  are  found  by  dividing  them  into  triangles. 
A  parabola  is  equal  to  3  of  its  circumscribing  parallelo- 

*  If  a  quadrilateral  can  be  inscribed  in  a  circle,  its  area 
•will  be,  (putting  s  semiperimeter,  and  a,  b,  c,  d,  the  sides,)  = 

\/s a  s—b.  s c.  s^d.    If  one  °f  the  sides,  as  d,  is  supposed  to 

vanish,  the  figure  merges  into  a  triangle,  and  the  formula  becomes 
V^H^  g^b.  T^c.  a.  That  is,  for  the  quadrilateral,  from  half 
the  sum  of  the  four  sides  subtract  each  side  separately:  multi- 
ply the  four  remainders  together;  the  square  root  will  be  the 
area.  For  the  triangle,  from  half  the  sum  of  the  three  sides 
subtract  each  side  separately;  multiply  the  three  remainders 
find  the  half  sum  together  ;  the  square  root  \vill  be  the  area. 


INSTRUMENTS  AND   THE   SLIDE-RULE.  125 

gram,  and  therefore  its  area  is  found  by  taking  §  of  the 
product  of  the  height  and  base.  A  circle  may  bo  con- 
ceived to  be  a  polygon  of  an  infinite  number  of  sides. 
Now,  a  polygon  is  made  up  of  as  many  triangles  as  the 
figure  has  sides,  and  the  area  of  each  triangle  is  found  by 
taking  the  product  of  the  height  and  half  the  basej  there- 
fore the  area  of  the  whole  polygon  will  be  equal  to  the 
perpendicular  multiplied  by  half  the  perimeter;  this, 
when  the  figure  merges  into  a  circle,  becomes  the  radius 
multiplied  by  half  the  circumference  j  or,  which  is  equiva- 
lent, the  diameter  multiplied  by  i  of  the  circumference. 
Now,  when  the  diameter  is  1,  the  circumference  is  3.1416, 
hence  1  X  .7854  =  the  area  of  a  circle  whose  diameter 
is  unity.  And  since  similar  surfaces  are  to  each  other  as 
the  squares  of  their  like  dimensions,  the  area  of  any  circle 
will  be  equal  to  the  square  of  its  diameter  multiplied  by 
.7854.  The  area  of  the  sector  of  a  circle,  in  like  manner, 
will  be  found  by  multiplying  the  radius  by  J  the  length 
of  the  arc.*  The  area  of  a  cycloid  is  3  times  that  of  its 
generating  circle.  From  the  method  described  in  page 
12,  of  projecting  the  circle  into  an  ellipse,  it  is  obvious 
that  the  area  will  be  in  proportion  to  the  elongation,  that 
is,  equal  to  the  product  of  the  axes  multiplied  by  .7854. 
The  sides  of  a  prism  being  parallelograms,  it  follows  that 
the  perimetrical  surface  will  be  equal  to  the  product  of 

*  To  find  the  length  of  the  arc,  from  8  times  the  chord  of  J 
the  arc  subtract  the  chord  of  the  whole  arc,  and  divide  by  3  ; 
the  quotient  is  the  length,  nearly. 

To  find  the  length  of  the  chord  of  J  the  arc,  add  together  the 
square  of  the  versed  sine,  or  height  of  the  segment,  and  the 
square  of  J  the  chord ;  the  square  root  is  the  length  of  the 
chord  of  .}  the  arc. 


11* 


126  A   TREATISE   ON   A   BOX   OF 

the  perimeter  and  height.  The  same  rule  applies  to  the 
cylinder,  which  is  a  round  prism.  The  sides  of  a  pyramid 
being  triangles,  the  area  of  each  of  which  is  found  by 
multiplying  J  the  base  by  the  height,  the  sloping  surface 
will  be  found  by  multiplying  J  the  perimeter  by  the 
slant  height.  The  same  will  be  the  case  with  the  cone, 
which  is  a  round  pyramid.  The  surface  of  a  sphere  is 
equal  to  the  convex  surface  of  its  circumscribing  cylinder, 
which  surface,  as  above  shown,  is  equal  to  the  circum- 
ference multiplied  by  the  depth ;  that  is,  in  this  case,  the 
circumference  multiplied  by  the  diameter.  The  same 
holds  good  for  the  surface  of  any  part  of  the  sphere,  it 
still  being  equal  to  the  surface  of  the  corresponding  paral- 
lel section  of  the  cylinder. 

We  may  now  refer  to  Table  5,  at  the  back  of  the  rule, 
the  gauge  points  of  which  are  determined  as  follows.  To 
reduce  square  inches  to  square  feet,  we  divide  by  144, 
and  -j/  144  =  12,  the  gauge  point  for  squares ;  144  -j- 
.7854  ==  183.3462,  and  j/ 183.3462  =  13.54  the  gauge 
point  for  circles,  when  the  dimensions  are  taken  in  inches, 
and  the  area  is  required  in  square  feet ;  and  so  of  the 
rest.  Putting  s  side,  b  base,^?  perimeter,  h  perpendicular 
height,  H  slant  height,  d  diameter,  c  conjugate  axis,  t 
transverse;  then 

Area  of  square  = Parallelogram  — 

square.  square. 

}U 


square.  square. 

Surface  of  prism  )  ph      Surface  of  pyra-  )  •_  _    \pll 

and  cylinder  )       square,     mid  and  cone   )       square. 


INSTRUMENTS   AND   THE    SLIDE-RULE.  127 

Area  of  circle  —  — — ; —  Ellipse  =   -: — r— 

circle.  circle. 

Area  of  cycloid  =  -^  Up-      Surface  of  sphere  =  -. — — 
circle.  circle. 

Surface  of  spherical  zone  = 

circle. 

That  is,  to  obtain  the  area  of  a  square,  over  the  square 
gauge  point  on  D,  set  1  of  the  slide,  then  over  the  side  on 
D  is  the  area;  for  the  parallelogram  over  the  square 
gauge  point  on  D,  set  the  base,  then  under  the  height  on 
A  is  the  area ;  or  find  a  mean  proportional  between  the 
base  and  height,  then  over  the  square  gauge  point  on  D 
set  1,  and  over  the  mean  proportional  on  D  will  be  the 
area ;  for  the  parabola,  over  the  square  gauge  point  on  D 
set  f  of  the  base,  then  under  the  height  on  A  is  the  area; 
for  the  surface  of  a  sphere,  over  the  circular  gauge  point 
on  D  set  4  of  the  slide,  then  over  the  diameter  on  D  is  the 
surface ;  and  so  of  the  rest. 

EXAMPLES. 

54.  The  diameter  of  a  circle  is  17  inches;  how  many 
square  feet  doe.-  it  contain  ? — Over  13.54  on  D  set  1  of 
the  slide,  over  17  Is  1.576  feet. 

55.  The  base  or  double  ordinate  of  a  parabola  is  39 
inches,  the  height  or  absciss  11.1  inches;   what  is  tho 
area  in  feet?— f  of  39  =  26.     Over  12  of  D  set  26  of 
the  slide,  under  11.1  of  A  is  2  feet. 

56.  The  diameters  of  an  ellipse  are  20  and  17  feet, 
what  is  the  area  in  square  yards? — Over  3.385  on  D  set 
20  on  the  slide,  under  17  on  A  is  29.67  yards;  or,  to  17 


128  A  TREATISE   ON   A   BOX   OP 

of  1)  set  17  of  the  slide,  under  20  is  18.44,  the  mean  pro- 
portional :  then,  over  3.385  on  D  set  1,  over  18.44  is 
29.67,  as  before. 

57.  The  side  of  a  square  measures  17  inches;  required 
the  area  in  square  feet. — Ans.  2  square  feet. 

58.  As  a  wheel,  5  feet  in  diameter,  is  rolled  along  by 
the  side  of  a  wall,  a  nail,  bent  sideways  over  the  tire, 
scratches  it,  and  marks  out  a  succession  of  curves,  termed 
cycloids ;  what  is  the  area  of  each  in  square  yards  ? — 
Ans.  6.545  yards. 

59.  A  circular  field  measures  283  yards  in  diameter; 
how  many  acres  does  it  contain  ? — Ans.  13  acres  nearly. 

60.  A  globe  is  7  feet  in  diameter ;  what  is  the  extent 
of  its  surface  in  square  yards? — Ans.  17.1. 

61.  A  piece  of  land  measures  95  links  by  74;  how  many 
square  perches  does  it  contain? — Ans.  11.24. 

62.  How  many  square  yards  of  canvas  will  be  required 
to  construct  a  conical  tent,  57  feet  round  the  bottom,  the 
slant  height  of  which  is  to  be  22  feet  ? — Ans.  69.7. 

63.  Kequired  the  surface,  in  square  feet,  of  a  pentago- 
nal prism,  the  length  168  inches,  and  each  side  of  the 
base  33  inches. — Ans.  192.5. 

64.  How  many  square  yards  are  contained  in  a  para- 
bola, of  which  the  base  is  126,  and  the  height  210  inches? 
—Ans   13.6. 

The  following  Table  of  Divisors  will  enable  the  student 
to  solve  the  preceding  questions  numerically.  The  same 
formulae  apply. 


INSTRUMENTS   AND   THE    SLIDE-RULE.  129 


Dimensions  in 

Area  in 

Square. 

Cii^le. 

Inches  

Sq.  Inches  

1 

1.2732 

Feet  

144 

183.3462 

Feet  

Yards  
Yards  

1296 
9 

1650.1164 
11.4591 

Rods  

272.25 

346.639 

Links  

Perches.... 

625 

795.7737 

Yards  

Perches.... 

30.25 

38.5154 

Acres  

4840 

6162.4719 

Chains  

Acres  

10 

12.7323 

Solution  of  Question  58  : 


3X5° 


=  6.545  square  yards. 


11.4591 

The  following  is  not  adapted  for  the  slide-rule ;  but  as 
it  is  an  excellent  method,  and  requisite  to  complete  the 
mensuration  of  surfaces,  it  is  accordingly  inserted. 

To  find  the  areas  of  plain  figures  by  an  odd  number  of 
equidistant  ordinates. 


Find  the  centre  of  the  figure  w,  and  draw  the  diameters 
rip,  dd.  On  each  side  of  w  set  off  any  equal  distances 
ws,  sv,  vo,  ?rr,  rt,  tm,  as  often  as  may  be  deemed  necessary, 
and  through  the  points  m,  t,  r,  &c.,  draw  the  ordinates  aa, 
bb,  cc,  &c.,  and  measure  their  lengths ;  also  the  distance 
nm,  or  op,  which  are  equal  to  each  other. 

Place  in  a  line  the  letters  x  4e  2o 

(Contractions  for  the  words,  extreme,  four  times  even,  twice  odd.) 

Sot  the  first  ordinate,  aa,  under  x ;  the  second,  bb,  un- 


J30  A   TREATISE   ON   A   BOX   OP 

der  4  e ;  the  third,  cc,  under  2  o ;  the  fourth  under  4  e ; 
the  fifth  under  2  o ;  the  sixth  under  4  e ;  and  so  on,  alter- 
nately, to  the  last,  which  set  under  x.  Add  up  the  three 
columns  separately.. 

Multiply  the  one  under  4  e  by  4 ;  and  the  one  under 
2  o  by  2. 

Add  the  three  together,  and  multiply  by  the  common 
distance  ws. 

For  the  end  areas  multiply  the  sum  of  the  extreme 
ordinates,  standing  under  x,  by  twice  the  height  nm ;  that 
is,  the  sum  of  the  bases  by  the  sum  of  the  heights. 

Add  this  product  to  the  other,  and  divide  by  3,  gives 
the  area. 

EXAMPLE. 

65.  In  a  curvilinear  figure,  7  ordinates  were  taken  in  the 
following  order,  20,  32,  38,  41,  39,  33,  22;  the  common 
distance  ws  was  8 ;  the  distance  nm  3  :  required  the  area — 
x  4  e  2  o 

"20        ~32        ~3i 
22        41        39 

~42       J®        77 

_6       106 

252        __f        154 

424 

154 
42 

620 

_8 

4960 

252 


1737i  area.  . 


INSTRUMENTS  AND   THE   SLIDE-RULE,  131 

The  greater  the  number  of  ordinates  taken,  the  more 
correct  will  be  the  area  found ;  and  when  the  curve  ana 
is  abrupt,  the  distance  nm  should  be  small.  If  the  curve 
taper  gradually  the  distance  nm  may  be  taken  equal  to 
mt,  and  then  the  extreme  ordinates  will  be  0.  The  num- 
ber of  ordinates  must  always  be  odd.  Beginning  with 
one  in  the  middle  insures  this. 

66.  In  a  curvilinear  figure  5  ordinates  were  taken,  70, 
79,  80,  78.6,69;  their  common  distance  was  24;  the 
height  of  each  of  the  end  areas  8  :  required  the  area. — 
Ans.  8176.53. 

67.  In  a  triangle  the  ordinates  were  0,  2,  4,  6,  8 ;  the 
common  distance  3;  required  the  area. — Ans.  48. 

68.  The  ordinates  in  a  triangle  are  0,  3,  6,  9,  12,  9,  6, 
3,  0;  the  common  distance  is  4;  what  is  the  area? — 
Ans.  192. 


SOLIDS. 

THE  next  part  on  the  Slide  Rule  is  Table  6,  which 
consists  of  a  number  of  gauge-points  for  the  mensuration 
of  Solids.  The  content  of  a  prism,  or  cylinder,  is  found 
by  multiplying  the  area  of  the  base  by  the  height.  Pyra- 
mids and  cones  are  J  of  their  circumscribing  prisms  and 
cylinders,  and  therefore  their  content  will  be  equal  to  the  pro- 
duct of  the  area  of  the  base,  and  £  of  their  height.  A  globe 
is  f  of  its  circumscribing  cylinder,  and  therefore  its  content 
is  equal  to  the  area  of  one  of  its  great  circles  multiplied  by 
f  of  its  diameter.  The  number  of  cubic  inches  in  a  gallon, 


132  A   TREATISE   ON   A   BOX   OF 

is  277.274 ;  hence,  if  the  dimensions  of  a  square  prism  are 
taken  in  inches,  and  the  content  is  required  in  gallons,  it 
will  be  (putting  I  the  length,  and  s  the  side  of  the  prism) 

Zs» 
277  274    "^  ^  277.274  =  16.65,  the  gauge  point  for 

square  prisms.  Since  the  pyramid  is  J  of  the  prism,  we 
may  multiply  by  the  whole  heiyht,  and  divide  by  three 

Is9 
times  277.274 :  that  is,  the  content  will  be  QQI  090  an<* 

I/  831.822  =  28.84,  the  gauge-point  for  square  pyramids. 
A  gallon  of  water  weighs  exactly  10  Ibs. ;  .-.  1  Ib.  occupies 
27.7274  cubic  inches;  dividing  this  by  the  specific  gravity 
of  any  metal,  and  taking  the  square  root  of  the  quotient, 
gives  the  gauge  point  for  such  metal.  The  gauge  points 
for  polygonal  prisms  are  obtained  by  dividing  the  number 
of  cubic  inches  in  a  gallon  by  the  polygonal  numbers  given 
at  page  118,  and  taking  the  square  root.  In  treating  of 
surfaces,  it  was  seen  that  the  area  of  a  circle,  inscribed  in 
a  square  whose  side  is  unity,  is  .7854.  Now,  277.274  -j- 
.7854  =  353.0353  ;  consequently,  the  dimensions  being 
taken  in  inches,  as  before,  the  content  of  a  cylinder  will  be 

I  da 
(putting  I  length  and  (^diameter)  ~      H-JKU  andy'SSS.OSSS 

=  18.78,  the  gauge  point  for  cylinders.  In  the  same  way 
as  the  pyramid  was  determined  by  taking  3  times  the 
prismatic  divisor,  so  the  content  of  the  cone  will  be  found 
by  taking  3  times  the  cylindrical  divisor;  and  3  X  353.0353 

Id* 
=  1059.106 ;   consequently,  content  =  -inrq  ing?  an<i 

I/ 1059.106  =  32.54  the  conical  gauge  point.  The  globe, 
being  $  of  the  cylinder,  will  be  twice  the  cone ;  hence  the 
divisor  will  be  the  half  of  1059.106,  namely,  529.553; 


INSTRUMENTS   AND   THE   SLIDE-RULE.  133 

therefore,  putting  d  the  diameter,  the  content  of  the  globe 

dd* 
will  be  590  553   and  -j/  529.553  =  23,  the  gauge  point 

for  globes.  By  having  divisors  and  gauge  points  thus 
prepared,  round  solids  are  reduced  to  square  ones,  by 
which  means  their  contents  are  determined  with  the 
greatest  ease,  as  they  all  come  under  the  general  formula 

Is" 

-p-j  in  which  I  represents  the  length,  s  the  side  or  diame- 
ter, as  the  case  may  be,  and  G  the  prepared  divisor,  or,  for 
the  purposes  of  the  Slide-rule,  its  square  root,  the  gauge 
point. 

For  finding  the  solidities  of  frustums  the  followifig  is 
an  invaluable  rule,  and  of  general  applicability  : — Find 
the  area  of  the  top,  the  area  of  the  bottom,  and  four  times 
the  middle  area ;  their  sum  is  six  times  a  mean  area,  which, 
being  multiplied  by  one-sixth  of  the  depth,  gives  the  con- 
tent. Now,  since  by  the  above-mentioned  divisors  we 
have  reduced  round  solids  to  square  ones,  the  rule  be- 
comes :  Add  together  the  square  of  the  top,  the  square  of 
the  bottom,  and  four  times  the  square  of  the  middle,  and 
multiply  the  sum  by  one-sixth  of  the  depth.  But  four 
times  the  square  of  a  number  is  equal  to  the  square  of 
twice  that  number ;  therefore  the  rule  becomes  still  easier. 
Add  together  the  square  of  the  top,  the  square  of  the  bot- 
tom, and  the  square  of  twice  tJie  middle,  and  multiply  by 
one-sixth  of  the  depth.  Moreover,  when  solids  do  not 
bulge  in  the  middle,  like  globes  and  spindles,  but  taper 
regularly  like  cones  and  pyramids,  then  the  sum  of  the 
top  and  bottom  will  l>e  twice  the  middle  diameter.  There- 
fore, for  all  regularly  tapering  frustums  the  above  given 
rule  becomes  still  more  concise,  viz.  :  Add  together  the 
12 


134  A  TREATISE   ON   A  BOX   Of 

square  of  the  top,  the  square  of  the  bottom,  and  the  square 
of  their  sum,  and  multiply  by  one-sixth  of  the  depth.  In 
the  same  way  as  for  cones,  we  multiplied  by  the  whole  of 
the  height,  and  took  three  times  the  divisor,  so  for  frus- 
tums we  may  take  the  whole  of  the  height,  and  divide  by 
six  times  the  divisor.  Now,  6  times  277.274  =  1663.644, 
whose  square  root  =  40.78,  the  gauge  point  for  square 
frustums.  Also,  6  times  353.0358  =2118.2148,  whose 
square  root  =  46,  the  gauge  point  for  round  frustums. 
Moreover,  as  a  rule  that  applies  to  frustums,  applies  also 
to  the  complete  solids  themselves,  and  as  this  is  of  such 
general  utility,  we  shall  illustrate  it  by  a  few  examples. 


EXAMPLES. 

69.  A  carpenter  has  a  block  of  wood  11  inches  square 
at  top,  13  inches  square  at  bottom,  and  12  inches  deep  : 
does  it  contain  an  exact  cubic  foot,  or  more,  or  less  ? 

Top  11"  ==  121 
Bottom  13a  =  169 
Sum  249  =  576 

866 

2  =  f  of  depth. 


1732  =  4  cubic  inches  more 
than  a  solid  foot. 

70.  A  prismoid,  24  inches  deep,  measures  12  inches 
by  10  at  top,  and  16  by  12  at  the  bottom;  what  is  the 
jontent  in  cubic  inches  ? 


INSTRUMENTS   AND   THE   SLIDE-RULE.  135 

Top  12  X  10  =  120 
Bottom  16  X  12  =  192 
Sum  28  X  22  =  616 


928  X  4  =  3712  cubic  inches. 


71.  A  wedge  measures  8  inches  along  the  edge;  the 
base  is  12  inches  long,  and  4  thick,  and  the  perpendicular 
height  18  inches;  what  is  the  solidity? 

Top         8X0=0 

Bottom  12  X  4  =  48 

Sum      20  X  4  =  80 

128  X  3  =  384  cubic  inches. 


72.  A  cylindroid,  or  solid  bounded  at  one  end  by  a 
circle  6  inches  diameter,  and  at  the  other  by  an  ellipse 
whose  axes  are  12  and  10,  is  24  inches  deep;  how  many 
gallons  will  it  contain  ? 

Top  6  X  6  =  36 
Bottom  12  X  10  =  120 
Sum  18  X  16  =  288 


444X4=1776; 

aod  =  5'03    alloPS- 


73.  What  is  the  solidity  of  a  globe  whose  diameter  is  1  ? 
See  diagram  page  48,  and  suppose  E  to  be  halfway 
between  A  and  F,  and  then  the  diameter  being  1,  A  C 

will  be  1,  and  AE  =  i 


136  A  TREATISE   ON  A   BOX   OP 

.•.  twice  EC  =  ~  v/3  the  middle  diameter;  then — 

Square  of  top  =  0 

Square  of  bottom  =  1 

Square  of  twice  middle  =  3 

4X  ^+.7854  =  .5236, 
the  content  as  determined  by  other  modes. 


To  return  to  the  slide.  Putting  I  or  Ti,  length,  height, 
Dr  depth,  according  as  the  solid  is  considered  lying  or 
standing ;  d  and  D}  less  and  greater  diameter ;  m,  middle 
diameter,  taken  halfway  between  them ;  r  and  R,  less  and 
greater  radius ;  s  and  S,  less  and  greater  side ;  g,  square 
root  of  product,  or  mean  proportional  between  two  dimen- 
sions; f}  fixed  axis;  and  v,  revolving  axis;  then  the  capa- 
cities of  solids  will  be  denoted  by  the  following  formulae : — 


1.  Prism  = 


prism 

; 

2.  Pyramid  = 


5.  Sphere  = 

globe 

fv* 

6.  Spheroid  =  -4-r— 

globe 


INSTRUMENTS   AND  THE   SLIDE-RULE.          137 

7_9 

7.  Rectangular  Prism  = 


8    Rectangular  Pyramid  = 


square  pnsrn 

¥ 


square  pyramid 

2AJ8 

9.  Parabolic  Prism  = — rj- 

square  pyramid 

10.  Elliptic  Cylinder  =    .ft*..   . 

cylinder 

11.  Elliptic  Cone  =  —£-• 

cone 

17   T^i 

12.  Paraboloid,  or  Parabolic  Conoid  =     ,.    ,    , 

cylinder 


or  -^-5  — 
cylinder 


13.  Hyperboloid,  or  Hyperbolic  Conoid 
h(D»  +  2m|«) 


~~  round  firustum'  globe 

?/>• 

14.  Parabolic  Spindle  =  ^^  X  -8 

15.  Spindles  in  general  = 

or       globe 

l(^+  ^9+*+  S\ 

16.  Fmstum  of  Pyramid  -  ^  frustum 


I  (da  +  D"  +  d  +  Z>|«) 
17.  Frustum  of  Cone  =  - 

round  frustum 


12* 


138  A   TREATISE   ON   A  BOX   OP 


18.  Frustum  of  Paraboloid  =-  —  , 

cylinder 


or 


cylinder 


19.  Frustum  of  Hyperboloid  =  ^^ 

round  frustum 


globe 

20.  Middle  Frustum  of  Parabolic  Spindle 

_  Z(<ff  -f  2  .  Z>9  —  ^  (of  2  diff.)8) 
cone 

21.  Middle  Frustum  of  Spindles  in  General 

_  7(<P  -f  D*  -f2in|«)         ?(r"  +  P*  4-  • 


round  frustum  globe 

22.  Middle  Frustum  or  Spheroid  = 


23.  Midddle  Frustum  of  Sphere  = 


cone 
~  *  of 


cylinder 
o 


cylinder 
24.  Any  Frustum  of  Sphere 


_ 

cylinder  cylinder 

25.  Segment  of  Sphere  =  -^  —  pr-^ 

On  examining  the  above  it  will  be  seen  that  the  formu- 
lae for  frustums  readily  resolve  themselves  into  those  for 
their  corresponding  complete  solids  ;  thus,  if  the  frustum 


INSTRUMENTS   AND   THE   SLIDE-RULE.  139 

in  formula  16  is  supposed  to  be  completed,  and  run  up  to  a 

7  f  ^3     I        O2  A 

point,  then  s  vanishes,  and  the  rule  becomes  — -^ J 

frustum 

and  since  the  frustum  divisor  is  double  the  pyramidical, 
the  numerator  and  denominator  cancel  by  2,  and  become 

IS* 

IT-    In  the  18th,  if  d  and  r  vanish,  the  formula  is 

pyramid 

resolved  into  the  12th,  and  so  of  the  rest.  A  pyramid,  as 
before  remarked,  is  equal  to  J,  a  parabolic  prism  to  §  of 
its  circumscribing  rectangular  prism.  A  cone  is  J,  a 
sphere  or  spheroid  f ,  a  paraboloid  £,  and  a  parabolic  spin- 
dle y£,  of  its  circumscribing  cylinder.  An  examination 
of  the  formulae  for  these  solids  will  show  that  they  are  so 
constructed.  Thus,  comparing  the  9th  with  the  8th,  we 
find  21  instead  of  1;  and  comparing  the  12th  with  the  3d> 
we  have  \Ji  for  h  or  I.  The  parabolic  spindle  being  T85 
of  the  cylinder,  and  the  cylinder  |  of  the  globe,  multiply- 
ing these  together  we  have  .8.  The  difference  between  an 
oblate  and  prolate  spheroid  will  be  best  understood  by 
considering  the  revolutions  of  a  parallelogram.  Suppose 
a  parallelogram  12  inches  by  6  to  be  divided  by  two  lines 
across  the  middle,  at  right  angles  to  each  other,  so  as  to 
cut  it  into  4  equal  portions,  each  6  by  3.  Then,  if  the 
parallelogram  revulve  on  the  short  axis,  it  will  generate  a 
cylinder  6  inches  deep,  and  having  a  diameter  of  12  inches; 
consequently,  its  content  will  be  6  X  12s  X  -7854.  If 
it  revolve  on  the  long  axis  the  cylinder  produced  will 
be  12  inches  deep,  and  6  diameter,  and  its  content 
12  X  6a  X  -7854 ;  that  is,  in  each  casefv*  .7854.*  Two- 

*  In  short,  all  the  formulae  for  round  solids  are  but  modifica- 

/w2 
tions  of  the  general  expression   ~-  ;  and  even  angular  solids 


140  A  TREATISE   ON   A  BOX   OF 

/»* 

thirds  of  this,  or  fv*  .5236,  gives  the  spheroid  =         - 

The  earth  is  a  spheroid  slightly  oblate,  the  polar  diameter, 
as  determined  by  careful  measurements  of  a  degree  at 
different  parts  of  its  surface,  being  about  26  miles  less 
than  the  equatorial,  the  prominence  of  the  torrid  zone 
having,  it  is  presumed,  been  acquired,  at  the  commence- 
ment, from  the  operation  of  centrifugal  force :  it  being 
supposed  that  the  earth  was  formed  from  matter  in  a  semi- 
fluid state,  and  set  rotating  on  its  axis  before  the  parts 
had  been  allowed  time  to  consolidate. 

ILLUSTRATION   OF  FORMULA. 

74.  Formula  1. — What  is  the  content,  in  gallons,  of  a 
vessel  in  the  shape  of  a  square  prism,  1  inch  deep,  and  29 
inches  along  each  of  the  sides  ? 

Referring  to  the  back  of  the  rule,  16.65  will  be  found 
the  gauge  point  for  square  prisms ;  therefore  over  16.65 
of  D  set  1 ;  over  29  is  3.03  gallons,  the  content.* 

75.  Formula  2. — Each  side  of  aii  hexagonal  pyramid  is 
46  inches,  its  perpendicular  depth  90  inches;  what  is  the 
content  in  gallons?     Over  17.9  set  90;  over  46  is  594.8 
gallons. 

76.  Formula  3. — The  depth  of  a  cylinder  is  40  inches, 

may  come  under  the  same  form,  if  we  conceive  them  to  be 
described  by  the  rotation  of  planes,  and  the  generated  surfaces 
subsequently  shaped  into  polygons  by  lateral  compression. 

*  Finding  the  content  of  solids  whose  depth  or  thickness  is 
unity  is  generally  termed  "  gauging  areas,"  because,  in  such 
cases,  the  surface  and  solidity  are  both  represented  by  the  same 
Dumber,  Is2  and  «2  being  equivalent  when  I  becomes  1. 


INSTRUMENTS   AND   THE   SLIDE-RULE.  141 

its  diameter  21.5;  how  many  gallons  will  it  contain? 
Over  18.78  set  40;  over  21.5  is  52.37  gallons. 

77.  Formula  4. — The  depth  of  a  cone  is  24  inches,  its 
diameter  17  ;  how  many  Ibs.  of  tallow  will  it  hold  ?    Over 
10.75  set  24;  over  17  is  60  Ibs. 

78.  Formula  5. — "What  is  the  weight  of  a  globe  of 
brass  8  inches  in  diameter?     Over  2.51  set  8;  over  8  is 
81.2  Ibs. 

79.  Formula  6. — The  fixed,  or  transverse  axis,  of  a 
prolate  spheroid  is  54  inches,  its  conjugate  33 ;  how  many 
bushels  will  it  contain?     Over  65.08  set  54;  over  33  is 
13.88  bushels. 

80.  The   fixed,  or   conjugate   diameter  of  an   oblate 
spheroid  is  33  inches,  its  transverse  54 ;  how  many  bushels 
will  it  contain?     Over  65.08  set  33;  over  54  is  22.7 
bushels. 

81 .  Formula  7. — A  cistern  in  the  shape  of  a  rectangulai 
prism,  or  parallelepiped,  is  82  inches  long,  54  broad,  and 
37.5  deep ;  how  many  gallons  will  it  contain  ?     To  54  on 
D  set  54  of  the  slide,  then  under  37.5  is  45,  a  mean  pro- 
portional.    Over  16.65  set  82;  over  45  is  598  J  gallons. 

82.  Formula  8. — A  vessel  oblong  at  top,  and  tapering 
downward  to  a  point,  measures   48  inches  by  75 ;    its 
depth  is  63  inches ;  how  many  Ibs.  of  hot  hard  soap  will 
it  hold?     Over  48  of  D  set  48,  then  under  75  is  60,  a 
mean   proportional.      Over    9.16   set   63;    over   60   is 
2700  Ibs. 

83.  Formula  9. — A  prismatic  vessel,  10  inches  deep, 
whose  ends  are  in  the  shape  of  a  parabola,  measures  80 


142  A   TREATISE   ON   A   BOX   OP 

inches  along  the  straight  side  or  double  ordinate,  from  the 
middle  of  which  to  the  vertex  is  60  inches ;  how  many 
gallons  will  it  contain?  Over  60  set  60,  under  80  ig 
69.28,  a  mean.  Over  28.84  set  20  (twice  depth;)  over 
69.28  is  115.4  gallons. 

84.  Formula  10. — The  axes  of  an  elliptic  cylinder  are 
67  and  52,  its  depth  50  inches ;  how  many  bushels  will 
it  contain  ?     Over  52  set  52 ;  under  67  is  59,  a  mean- 
Over  53.14  set  50;  over  59  is  61.6  bushels. 

85.  Formula  11. — The  axes  of  an  inverted  elliptic  cone 
are  16  and  9,  the  depth  19  inches ;  how  many  pints  will 
it  hold  ?     Over  16  set  16 ;  under  9  is  12,  a  mean.    Over 
11.5  set  19  ;  over  12  is  20.6  pints. 

86.  Formula  12. — A  vessel  in  the  shape  of  a  parabolic 
conoid  is  42  inches  deep,  and  the  diameter  of  the  top  is 
24  inches;  what  is  the  content  in  gallons  ?     Over  18.78 
set  21  (half  42;)  over  24  is  34.25  gallons:  or  by  the 
second,  over  18.78  set  84  (twice  42);  then  over  12  is 
34.25,  as  before. 

87.  Formula  13. — "What  is  the  content,  in  gallons,  of 
a  hyperbolic  conoid,  the  diameter  at  top  being  52  inches, 
the  diameter  in  the  middle  34,  and  the  depth  25  inches  ? 
Over  46  set  25,  then- 
Over  52  is  31.9 

Over  68  is  54.6 


86.5  gallons. 

88.  Formula  14. — What  is  the  content,  in  gallons,  of 
a  parabolic  spindle,  the  diameter  of  which  is  28  inches, 


INSTRUMENTS   AND  THE   SLIDE-RULE.  143 

and  length  70  inches?  Over  23  set  70;  over  28  is 
103.7.  Then  to  103.7  on  A  set  commencement  of  slide, 
and  over  .8  is  82.96  gallons. 

89.  Formula  15. — The  length  cf  a  spindle  is  20  inches, 
the  greatest  diameter  6  inches,  and  the  diameter  halfway 
between  it  and  the  point  4.74  inches ;  what  is  the  content 
in  cubic  inches  ?  Over  2. 76  set  20,  then — 

Over  6.       is    94.       . 
Over  9.48  is  235.5 


329.5  cubic  inches. 


90.  Formula  16. — How  many  gallons  will  be  contained 
in  the  frustum  of  an  octagonal  pyramid,  each  side  of  the 
greater  base  being  17.5  inches,  of  the  less  14  inches,  and 
the  perpendicular  depth  47  inches?  Over  18.55  set  47, 
then — 

Over  14.    is    26.8 

Over  17.5  is    41.8 

Over  31.5  is  135.2 


203.8  gallons. 

91.  How  many  Ibs.  of  hot  hard  soap  will  the  above 
contain  ? 

As  polygonal  pyramids  are  not  figures  of  frequent  oc- 
currence, it  was  not  deemed  necessary  to  insert  gauge 
points  for  any  other  quantities  than  gallons  and  cubic  feet, 
the  weight  therefore  must  be  determined  by  a  second  pro- 
cess, which,  since  a  gallon  of  water  weighs  10  Ibs.,  is  ef- 
fected by  multiplying  the  content  in  gallons  by  10  times 


144  A  TREATISE   ON   A  BOX   Of 

the  specific  gravity.  Now  the  specific  gravity  of  hot  hard 
soap  is  shown  on  the  rule  to  be  .99,  ten  times  which 
:=9.9.  Therefore,  to  203.8  on  A  set  commencement  of 
slide,  then  over  9.9  is  2017  Ibs. 

92.  Formula  17. — The  frustum  of  a  cone  is  43  inches 
deep,  the  diameter  at  one  end  36,  at  the  other  20  inches  • 
how  many  bushels  will  it  contain  ?  Over  130.17  set  43; 
then — 

Over  20  is  1.02 

Over  36  is  3.28 

Over  56  is  7.96 


12.26  bushels.* 


93.  Formula  18. — The  diameters  of  the  frustum  of  a 
paraboloid  are  30  and  40  inches,  the  depth  1 8  inches ; 
how  many  gallons  will  it  contain  ?     Over  18.78   set  9 
(half  18,)  then- 
Over  30  is  23. 

Over  40  is  40.7 

63.7  gallons. f 

94.  Formula  19. — How  many  bushels  will  be  contained 
in  the  frustum  of  a  hyperbolic  conoid,  the  top  and  bottom 

*  If  the  frustums  of  two  equal  cones  be  joined  together  at  their 
greater  ends  they  form  a  figure  called  by  gaugers  a  cask  of  the 
4th  variety. 

flf  the  frustums  of  two  equal  paraboloids  be  joined  together 
at  their  greater  ends  they  form  a  figure  called  by  gaugers  a  cask 
of  the  3d  variety. 


INSTRUMENTS   AND   THE   SLIDE-RULE.  145 

diameters  of  which  are  23  and  40  inches,  the  middle  36 
inches,  and  depth  20  inches?  Over  130.17  set  20, 
then — 

Over  23  is     .62 

Over  40  is  1.88 

Over  72  is  6.13 

8.63  bushels. 

95.  Formula  20. — The  length  of  a  vessel  in  the  form 
of  the  middle  frustum  of  a  parabolic  spindle  is  20,  the 
greatest  diameter  16,  and  least  12  inches;  what  is  the 
content  in  gallons  ?  Here  twice  the  difference  of  the 
diameters  =  8 ;  therefore,  over  32.54  set  20,  then — 

Over  12  is  2.72 

Over  16  is  4.83 

4.83 


12.38 
Over  8  is  1.2,  one-tenth  of  which  is          .12 

12.26  gallons.* 

96.  Formula  21. — The  bung  diameter  of  a  vessel  is  36 
inches,  the  head  30,  twice  the  diameter  taken  midway 
between  them  67.8  inches,  and  the  length  40  inches ;  how 
many  gallons  will  it  contain  ?     Over  46  set  40,  then — 
Over  30      is  17. 
Over  36      is  24.5 
Over  67.8  is  86.9 


128.4  gallons. 


*  A  cask  in  the  form  of  the  middle  frustum  of  a  parabolic 
spindle  is  termed  by  gangers  a  cask  of  the  2d  variety. 

13 


146  A   TREATISE   ON   A   BOX   OP 

97.  Formula  22. — A  vessel  in  the  form  of  the  middle 
frustum  of  a  prolate  spheroid  is  40  inches  long,  the  bung 
diameter  is  36,  and  the  head  27  inches ;  what  is  the  con- 
tent in  gallons  ?  Over  32.54  set  40,  then — 

Over  27  is  27.4 
Over  36  is  49. 
49. 


125.4  gallons.* 

98.  Formula  23. — How  many  cubic  feet  are  contained 
in  the  middle  zone  of  a  sphere,  the  axis  of  which  is  44 
inches,  and  the  height  of  the  zone  14  inches?  Over  46.9 
set  14,  then — 

Over  44  is  12.32 
Over  14  is  1.26,  one-third  of  which   =   .42 

11.9  bushels. 


99.  Formula  24. — What  is  the  content  in  gallons  of  the 
shoulder  of  a  still  in  the  form  of  the  frustum  of  a  sphere, 
the  top  and  bottom  diameters  being  42  and  36  inches,  and 
the  height  30  inches?  Over  18.78  set  15  (half  30,) 
then — 

Over  36  is  55 
Over  42  is  75 
Over  30  is  38.3 
+  *  =  12.7 


181.    gallons. 


*  A  cask  in  the  form  of  the  middle  frustum  of  a  prolate  sphe- 
roid is  termed  by  gaugers  a  cask  of  the  1st  variety 


INSTRUMENTS   AND   THE   SLIDE-RULE.  147 

100.  Formula  25. — A  copper  basin  in  the  form  of  the 
segment  of  a  sphere  is  18  inches  deep,  the  diameter  across 
the  top  40  inches ;  how  many  gallons  will  it  contain  f 
Over  23  set  18,  then — 

Over  18  is  11 
Over  20  is  13.6 
+  twice  ditto  =  27.2 

51.8  gallons. 

The  content  of  cylindroids,  prismoids,  and  wedges,  is 
found  by  taking  the  mean  proportionals  of  the  products 
of  the  top  and  bottom  dimensions,  and  of  the  product  of 
their  sums,  making  use  of  the  round  frustum  gauge  points 
for  the  cylindroid,  and  the  square  frustum  for  the  prismoid 
and  wedge. 

ILLUSTRATION. 

101.  The  perpendicular  depth  of  a  cylindroid  is  52 
inches,  the  diameters  at  top  60  and  46,  at  bottom  42 
inches ;  what  is  the  content  in  bushels  ? 

Bottom  42  X  42,  mean  proportional  between  which  =42 
Top      60x46  "  "  =52.54 

Sum    102X88  "  «          *=94.74 

Over  130.17  set  52,  then  over  42      =    5.4 

52.54=    8.5 
94.74  =  27.5 

41.4  bushels. 

Referring  to  the  Table  on  page  150,  the  round  frustum 
divisor  for  bushels  is  16945.74. 


148  A   TREATISE   ON  A  BOX  OP 

The  numerical  solution  of  this  question,  therefore,  will 
be  as  follows : — 

Bottom  42  X  42  =  1764 
Top  60  X  46  =  2760 
Sum  102  X  88  =8976 

13500 
52 


27000 
67500 


16945.74  )  702000.00  (  41.426  bushels. 
6778296 

2417040 
1694574 


7224660 
6778296 

4463640 
3389148 


577476 

For  questions  102  and  103  the  divisor,  as  shown  on 
page  150,  will  be  13309.15. 

102.  The  length  and  breadth  of  a  coal  wagon  at  top 
are  81  and  55  inches,  at  bottom  41  and  29  inches;  the 
depth  is  47  inches ;  how  many  bushels  will  it  contain  ? 
Top       81 X55,  mean  proportional  between  which  =  66.8 
Bottom  41x29  "  "  =  35.5 

Sum    122X84  "  «  =101.2* 

*  It  must  be  observed  that,  in  irregular  solids,  the  menn  pro- 
portional of  the  sum  is  not  the  sum  of  the  mean  proportionals ; 


INSTRUMENTS  AND   THE   SLIDE-RULE.  149 

Over  115.3  set  47,  then  over  66.8  =  15.6 

35.5=   4.4 
101.2  =  36.1 


56.1  bushels. 


103.  A  heap  of  malt  is  piled  into  the  form  of  a  wedge, 
24  inches  deep,  the  base  is  40  inches  long,  and  20  broad, 
the  edge  20  inches  long;  how  many  bushels  does  it 
contain  ? 

Top       20  X  0 

Bottom  30x20,  mean  proportional  between  which = 28. 28 

Sum      60X20,  "  "  =34.37 

Over  115.3  set  24,  then  over  28.28  =  1.46 

34.36=2.14 


3.6    bushels. 


the  former  mast  be  taken,  not  the  latter.  In  prismoids,  also, 
attention  must  be  paid  to  the  position  of  the  sides  ;  for  the  top 
and  bottom  areas  of  two  prismoids  may  be  the  same,  and  yet 
their  middle  area,  and  consequently  their  content,  different. 
For,  suppose  a  prismoid  to  be  12  inches  by  10  at  top,  and  9 
inches  by  6  at  bottom,  the  12  falling  over  the  9,  and  the  10  over 
the  6  :  if  now  we  shift  the  position  of  the  top  parallelogram  so 
as  to  bring  the  short  side  over  the  long  one  of  the  bottom,  then 
the  figure  becomes  distorted,  and  the  content  altogether  altered. 

In  the  first  it  will  be  —  In  the  second  — 

Bottom  12  X  10  =  120  Bottom  12  x  10  =  120 

Top          9  X    6  =    54  Top         6  X    9  =    54 

Sum      21  X  16  =  336  Sum       18  X  19  =  342 


510  XL  516  X 

fa  o 


13* 


150 


A   TREATISE   ON   A   BOX   OF 


The  following  Table  of  Divisors  will  enable  the  Student  to  solve  the 

preceding  Questions  numerically.     The  same  formulae  apply. 

DIMENSIONS  IN 
INCHES. 

V 

SQUARE  SOLIDS. 

ROUND  SOLIDS. 

Content  in 

£& 

Prism. 

Pyramid. 

Frustum. 

Cylinder. 

Globe. 

Cone. 

Frustum. 

Cubic  Inches    . 

1. 

3. 

6. 

1.2732 

1.90985 

3.8197 

7.6394 

Cubic  Feet    . 

1728. 

5184. 

10368. 

2200.16 

3301 

.24 

6600.4 

8 

13200.96 

Pints  . 

34.659 

103.978 

207.956 

44.129 

66. 

194 

132.38 

3 

264.776 

Gallons  . 

277.274 

831.82 

1663.64 

353.03(5 

529. 

554 

1059.1 

)8 

2118.216 

Bushels     . 

2218.19 

6654.57 

13309.15 

2824.29 

4236 

435 

8472.8 

7 

16945.74 

Hot    hard      \ 
Soap,  Ibs.  '  J 

.99 

28. 

84. 

168. 

35.65 

53.475 

106.95 

213.9 

Cold  ditto  . 

1.02 

27.14 

81.42 

162.84 

3455 

51.8 

32 

103.66 

4 

207.327 

Tallow  . 

.915 

30.28 

90.84 

181.68 

3855 

57. 

83 

115.65 

231.3 

Flint  Glass 

3.21 

8.64 

25.92 

51.84 

11 

16. 

5 

33. 

66. 

Plate  ditto     . 

2.418 

11.3 

33.9 

67.8 

14.4 

21. 

6 

43.2 

86.4 

Platinum  . 

21.45 

1.29 

3.87 

7.74 

1.642 

2.4 

63 

4.927 

9.854 

Gold      . 

19.25 

1.44 

4.32 

8.64 

18 

2.7 

5 

5.5 

11. 

Mercury   . 

13.61 

2.03 

6.09 

12.18 

2.584 

3.8 

77 

7-753 

15.5 

Lead 

11.35 

2.44 

7.32 

14.64 

3.11 

4.6 

7 

9.34 

18.68 

Silver 

10.53 

2.64 

7.92 

15.84 

3.36 

5.0 

4 

10.08 

20.16 

Copper  . 

8.81 

3.155 

9.46 

18.92 

4. 

6. 

12. 

24. 

Brass  . 

8.41 

3.3 

9.91 

19.82 

4.2 

6.3 

12.6 

25.2 

Wt.  Iron  &  Steel 

7.82 

3.54 

10.62 

21.24 

4.5 

6.7 

5 

13.5 

27. 

Ct.  Iron,  Tin,  1 
&  Zinc  .     J 

7.24 

3.8 

11.41 

22.82 

4.8 

7.2 

14.44 

28.8 

Tee  &  Gunpowd. 

.93 

29.81 

89.43 

179.86 

37.94 

56.91 

113.82 

227.64 

POLYGONAL  SOLIDS. 

CONTENT  IN  GALLONS. 

CONTENT  IN  CUBIC  FEET. 

DIMENSIONS  IN 

INCHES. 

Priam. 

Pyramid. 

Frustum. 

Prism. 

Pyramid. 

Frustum. 

Trigonal 

640.34 

1921.01 

3842.02 

3090.64 

11971.93 

23943.87 

Tetragonal 

277.274 

831.82 

1683.64 

1728. 

5184. 

10368. 

Pentagonal     . 

161.161 

483.48 

966.96 

1004.37 

3013.12 

6026.23 

Hexagonal  .       . 

106.72 

320.16 

640.32 

665.11 

1995.32 

3990.64 

Heptagonal    . 

76.39 

229.17 

458.34 

475.52 

1426.56 

2853.12 

Octagonal    . 

57.42 

172.27 

344.54 

357.88 

1073.64 

2147.28 

Nonagonal     . 

44.85 

134.56 

269.12 

279.53 

838.58 

1677.17 

Decagonal  . 

36.03 

108.11 

216.22 

224.58 

67375 

1347.50 

Undecagonal  . 

29.6 

88.81 

177.63 

184.5 

553.51 

1107.02 

Dodecagonal 

24.76 

74.30 

148.6 

154.34 

463.02 

926.03 

8* 

Numerical  solution  of  Question  78.            —  =81.2  Ibs. 
6.3 

Numerical  solution  of  Question  90. 

Top        14.8    —196 

Bottom  17.52  =  306.25 

Sum       31.52  _  992.25 

70241.5 

1494.5  X  47  —  70241.5  ;  and                = 
344.54 

203.8  gallons. 

INSTRUMENTS  AND   THE   SLIDE-RULE.  151 


EXAMPLES   FOR   PRACTICE. 

104.  What  is  the  weight  of  a  prism  of  steel  7  inches 
square,  15  inches  long? — Ans.  20T  Ibs. 

105.  What  would  be  the  weight  of  a  pyramid  of  ice, 
8  inches  square  at  bottom,  and  13.8  inches  high  ? — Ans. 
9.87  Ibs. 

106.  A  cylindrical  glass-pot,  24  inches  diameter,  is 
charged  with  flint  glass  to  the  depth  of  15  inches :  what 
is  its  weight  ? — Ans.  785  Ibs. 

107.  An  inverted  cone  is  23  inches  deep,  its  diameter 
at  top  10  inches  :  what  quantity  of  tallow  will  it  contain  ? 
—Ans.  19.8  Ibs. 

108.  What  quantity  of  gunpowder,  shaken  down,  will 
fill  a  shell  whose  internal  diameter  is  9  inches? — Ans. 
12.8  Ibs. 

109.  The  axes  of  an  oblong  or  prolate  spheroid  are  6 
and  8  inches  :  what  quantity  of  mercury  will  it  contain  ? 
—Ans.  74.2  Ibs. 

110.  The  axes  of  an  oblate  spheroid  are  6  and  8  inches : 
what  quantity  of  mercury  will  it  contain  ? — :Ans.  99  Ibs. 

111.  What  is  the  weight  of  a  rectangular  block  of 
ice,  12  inches  by  10  thick,  and  30  inches  long  ? — Ans. 
121  Ibs. 

112.  The  top  of  an  inverted  rectangular  pyramid  mea- 
sures 17  inches  by  13 ;  its  depth  is  44  inches  :  how  many 
gallons  of  water  will  it  contain,  and  how  many  Ibs.  ? — • 
Ans.  11.69  gallons,  116.9  Ibs. 


152  A   TREATISE  ON   A  BOX  OP 

113.  The  base  of  a  parabola  is  32  inches,  its  absciss  24 
inches ;  the  depth  b  1  inch  :   how  many  gallons  will  it 
hold  ? — Ans.  1.84  gallons. 

114.  The  diameters  of  an  elliptic  cylinder  are  25  and 
20  inches,  the  depth  13  inches :  how  many  gallons  will 
it  contain? — Ans.  18.4  gallons. 

115.  An  elliptic  cone  of  silver  is  10  inches  high,  the 
diameters  at  bottom  5  inches  by  4  :  what  is  its  weight  in 
Ibs.  avoirdupois? — Ans.  19.8  Ibs. 

116.  A  paraboloid  of  copper  is   12  inches  high,  the 
diameter  of  the  base  8  inches :   what  is  its  weight  ? — 
Ans.  96  Ibs. 

117.  A  vessel  in  the  shape  of  an  hyperboloid  is  25 
inches  deep,  the  radius  of  the  top  26,  and  the  middle 
diameter  34  inches :  what  quantity  of  cold  hard  soap  will 
it  hold?— Ans.  883  Ibs. 

118.  The  length  of  a  parabolic  spindle  is  82  inches,  its 
diameter  10  inches :  required  the  content  in  gallons  — 
Ans.  4.83  gallons. 

119.  The  length  of  a  cast-iron  spindle  is  20  inches,  its 
greatest  diameter  9  inches,  and  the  diameter  halfway  be- 
tween that  and  the  point  6  inches :  what  is  its  weight  ? — 
Ans.  155.8  Ibs. 

120.  The  frustum  of  a  nonagonal  pyramid,  25  inches 
deep,  measures  9  inches  along  each  side  at  top,  and  12  at 
bottom:  how  many  gallons  will  it  contain? — Ans.  61.8 
gallons. 

121.  Suppose  a  cask  to  consist  of  two  equal  frustums 
of  a  cone,  the  length  of  which  is  40  inches,  the  bung  dia- 


INSTRUMENTS  AND   THE   SLIDE-RULE.  153 

meter  32,  and  the  head  24  :  what  is  the  content  in  gallons? 
By  formula  17. — Ans.  89.4  gallons,  4th  variety. 

122.  Suppose  a  cask,  of  the  same  dimensions,  to  be 
composed  of   two    equal   frustums  of  a  paraboloid :    re- 
quired the  content.     By  formula  18. — Ans.  90.6  gallons, 
3d  variety. 

123.  Let  the  cask  be  the  middle  frustum  of  a  para- 
bolic spindle,  and  the  dimensions  remain  the  same  :  what 
is  the  content?      By  formula  20. — Ans.  98.2  gallons, 
2d  variety. 

124.  Let  the  cask  be  the  middle  frustum  of  a  prolate 
spheroid,  the  dimensions  continuing  the  same :    what  is 
the  content?     By  formula  22. — Ans.  99.1  gallons,  1st 
variety. 

125.  What  will  be  the  content  of  the  middle  frustum 
of  a  spindle  having  the  same  dimensions,  and  also  the 
diameter   halfway   between    the    head    and    bung    29.6 
inches?     By  formula  21. — Ans.  96.45  gallons,  true  con- 
tent. 

126.  Required  the  content  in  cubic  feet  of  the  middle 
frustum  of  a  sphere,  the  height  of  which  is  24,  and  the 
least  diameter  18  inches. — Ans.  7.72  feet. 

127.  Find  the  content  in  gallons  of  the  frustum  of  a 
sphere,  the  height  of  which  is  9  inches,  and  the  radii  at 
its  ends  14  and  10  inches. — Ans.  16.47  gallons. 

128.  What  is  the  weight  of  the  segment  of  a  globe  of 
leal,  the  height  of  which  is  6  inches,  and  the  radius  of 
the  base  8  inches  ? — Ans.  293  Ibs. 

129.  The  depth  of  a  cylindroid  is  50  inches,  the  diame- 


154  A  TREATISE  ON  A   BOX   OP 

ters  of  the  elliptic  base  are  60  and  44  inches,  the  diameter 
of  the  circular  top  40  inches :  required  the  content  in 
gallons. — Ans.  298.3  gallons. 

130.  The  depth  of  a  prismoid  is  50  inches ;  the  base  is 
a  parallelogram  60  inches  long,  44  broad ;   the  top  is  a 
square,  the  sides  of  which  are  each  40  inches :  what  is 
the  content  in  gallons? — Ans.  379.8  gallons. 

131.  The  frustum  of  a  square  pyramid  is  30  inches 
deep,  each  of  the  sides  at  bottom  36,  and  at  top  25 
inches :  what  is  the  content  of  each  of  the  wedges  into 
which  a  diagonal  plane,  passing  through  its  extremities, 
divides  it? — Ans.  62.97  gallons,  lower  hoof  or  wedge; 
38.77  ditto,  upper  ditto. 


TABLES  VII.  and  VIII.,  at  thi  back  of  the  rule,  are 
adapted  for  the  use  of  the  E  slide.  Table  VII.  exhibits 
the  weight  of  metallic  spheres,  commencing  thus : — 

D  diameter.          E  weight. 
Platinum  4    inches  =    26  Ibs.  avoirdupois. 
Gk)ld         6*  inches  =  100  Ibs. 

That  is,  over  4  on  D,  set  26  Ibs.  on  E,  then  the  numbers 
on  D  will  be  a  series  of  diameters,  and  the  numbers  over 
them  on  E  their  corresponding  weights. 

EXAMPLES. 

132.  A  sphere  of  platinum  weighs  51  Ibs. :  what  is  its 
diameter? — Over  4  set  26  Ibs.;  under  51  Ibs.  is  5  inches. 

133.  A  sphere  of  silver  weighs  7  Ibs.:   what  is  its 
diameter? — Ans.  3.284  inches. 


INSTRUMENTS   AND  THE   SLIDE-RULE.  155 

134.  Rockets  receive  their  names  from  a  comparison 
of  the  external  diameters  of  their  cases  with  leaden  balls : 
what,  then,  is  the  diameter  of  a  5-pound  rocket  ? — Ana. 
2.86  inches. 

135.  A  globe  of  wrought  iron  weighs  19.7  Ibs. :  what 
is  its  diameter? — Ans.  5.1  inches. 

136.  A  spherical  vessel,  filled  with  mercury,  holds  258 
Ibs. :  what  is  its  diameter  ? — Ans.  10  inches. 

137.  Thirteen  Ibs.  of  gunpowder  fill  a  shell:  what  is 
its  diameter? — Ans.  9.04  inches. 

138.  A  sphere  of  brass  weighs  81.2  Ibs. :  what  is  its 
diameter  ? — Ans.  8  inches. 


Table  VIII.  is  used  precisely  like  Table  VII.,  and  is 
for  finding  the  diameters  and  circumferences  of  spheres 
from  their  solidities;  and  also  the  solidities  of  regular 
bodies,  the  tetrahedron,  &c. 

EXAMPLES. 

139.  The  solidity  of  a  sphere  is  33.6 :  what  is  its 
diameter  ?— Over  4.6  of  D  set  51  of  E;  under  33.6  is  4. 

140.  A  globe  contains  98.5  solid  feet :  what  is  its  cir- 
cumference?— Ans.  18  feet. 

141.  The  side  of  a  tetrahedron  measures  2.2  inches: 
how  many  cubic  inches  does  it  contain? — Ans.  1.25  cubic 
inches. 

142.  The  side  of  an  octahedron  measures  3.3  inches : 
how  many  cubic  inches  does  it  contain? — Ans.  16.875 
solid  inches. 


156  A   TREATISE   ON   A   BOX   OF 

143.  A  dodecahedron  contains  15  cubic  feet:  what  is 
the  length  of  each  of  its  sides  ? — Ans.  1.25  feet. 

144.  The  solidity  of  an  icosahedron  is  162  :  what  is 
the  length  of  each  of  its  sides  ? — Ans.  4.2. 


SOLAR  SYSTEM. 

THE  concluding  part  of  the  tabular  work  on  the  rule  is 
for  the  use  of  the  line  A  in  conjunction  with  that  of  E. 
According  to  Kepler's  famous  discovery,  the  squares  of 
the  periodic  times  of  the  planets  are  proportional  to  the 
cubes  of  their  mean  distances.  Now,  since  the  line  A  is 
laid  down  twice,  and  the  line  E  thrice,  in  the  same  spa.ce, 
when  the  slide  E  is  laid  evenly  in,  the  cubes  of  the  num- 
bers on  A  will  be  equal  to  the  squares  of  the  numbers  on 
E ;  when  in  any  other  position,  the  cubes  of  the  numbers 
on  A  will  be  proportional  to  the  squares  of  the  numbers 
on  E.  Hence,  if  under  95  millions  of  miles  on  A  we 
set  365  days,  or  52  weeks,  or  13  lunar  months,  or  1 
year,  on  E ;  then  the  numbers  on  A  will  be  a  series  of 
planetary  distances,  and  the  numbers  beneath  them  on  E 
their  periods  of  revolution,  in  days,  weeks,  months,  or 
years,  according  as  365,  52,  13,  or  1,  is  selected. 

EXAMPLES. 

145.  The  distance  of  Mercury  from  the  sun  is  37  mil- 
lions of  miles;  what  is  the  length  of  his  year? — Under  95 
set  365 ;  under  37  is  88  days. 

146.  Mars  is  about  687  days  in  revolving  round  the 
sun ;  what  is  his  distance  ? — Ans.  144  millions. 

147.  Herschel's  mean  distance  is  about  1823  millions 
of  miles ;  how  many  years  does  he  consume  in  traversing 
his  orbit? — Ans.  83.8  years. 


INSTRUMENTS  AND   THE   SLIDE-RULE.  157 

148.  If  a  planet  revolved  in  an  orbit  20  million  miles 
from  the  sun ;  how  long  would  it  take  in  passing  round 
him? — Ans.  35 J  days. 

149.  Suppose  the  recently  discovered  planet  to  be  3,000 
millions  of  miles  distant  from  the  sun ;  how  many  years 
does  it  take  in  traversing  its  orbit  ? — Ans.  178  years. 

150.  The  nearest  of  Saturn's  moons  is  108  thousand 
miles  distant  from  him,  and  the  time  of  its  periodic  revo- 
lution about  22  J  hours;  the  second  is  distant  140  thou- 
sand :  what  is  its  periodic  revolution  ? — Under  108  of  A 
set  22  J  of  E;  under  140  is  34  hours  nearly. 

151.  The  fourth  satellite  of  Saturn  spends  65  hours  in 
passing  round  its  primary;    required  its  distance  from  ' 
him.— Ans.  217,000  miles. 

The  following  table  will  enable  the  student  to  solve 
the  previous  questions  numerically : — 

WEIGHT    OF    METALLIC    SPHERES. 

Platinum 4     inches  diam.  =  26  Ibs.  av. 

Gold 6.5  ...  =100 

Mercury 3  ...  =  7 

Lead 7.5         ...  =  90 

Silver 4.6          ...  =  18 

Copper 6  ...  =  36 

Brass 5.4          ...  =  25 

Wt  Iron  and  Steel 3  ...  =  4 

Ct.  Iron,  Tin,  and  Zinc  6  ...  =  30 

Ice  and  Gunpowder  ....7  ...  =  6 

Solidity  of  Sphere  =  .5236  d*  =  .01688  e». 
Solidity  of  Tetrahedron     =    .11785  s1. 

"          Octahedron       =    .47140  a1. 

"          Dodecahedron  =  7.66312  «». 

'•         Icosahedron     =  2.18169  •* 


158  A   TREATISE   ON   A   BOX   OF 

Numerical  Solution  of  Question  133. 

lb.  3  lb. 

18  :  4.5  :  :       7  :  d3;  or  taking  the  j<Kof  each  term 

4.5  3/7 

^K18  :  4.5  :  :  ^K7  :  • '     * — ;  which,  multiplying  nu- 
merator and  denominator  by  jjK18a  =    '    ^ '       -    =: 
k  ^2268  =  3.284. 
Question  148. 


and  ^/1243  =  35.25  days. 


MISCELLANEOUS  QUESTIONS. 

152.  A  triangular  piece  of  board,  measuring  18  feet  in 
perpendicular  height,  is  to  be  divided  equally  among  4 
men,  by  sections  parallel  to  the  base :  at  what  distance 
from  the  vertex  must  they  be  cut  ? — Similar  surfaces  vary 
as  their  squares ;  hence,  over  18  of  D,  set  4  shares  on  C ; 
then  under  3  shares  is  15.58  feet;  under  2  is  12.72,  and 
under  1  is  9  feet. 

153.  A  circle  measures  9  inches  in  diameter  :  required 
the  diameter  of  another  of  twice  the  area. — Over  9  of  D, 
setl;  under  2  is  12.72. 

154.  Four  men  bought  a  grindstone,  30  inches  in  dia- 
meter, and  agreed  that  the  first  should  use  it  till  he  ground 
down  }  of  it  for  his  share,  deducting  6  inches  in  the  mid- 
dle for  waste ;  then,  that  the  second  should  use  it  till  he 


INSTRUMENTS   AND  THE   SLIDE-RULE.  159 

ground  down  }  part,  and  so  on  :  what  part  of  the  dia- 
meter must  each  grind  down  ? — If  £th  of  the  diameter  be 
waste,  2'5th  of  the  content  is  waste;  therefore,  conceiving 
the  whole  to  contain  25  shares,  1  share  will  be  waste,  and 
each  man  will  have  6  shares.  Over  6  inches  on  D,  set  1 
share  on  C ;  under  7  is  15.87  ;  under  13  is  21.63 ;  under 
19  is  26.15;  and  under  25  is  30.  Subtracting  these 
numbers  from  each  succeeding  one,  we  obtain  9.87  inches 
for  the  fourth;  5.76  for  the  third;  4.52  for  the  second; 
and  3.85  for  the  first. 

155.  Three  persons  having  bought  a  sugar-loaf,  20 
inches    high,  it  is  required  to  divide  it  equally  among 
them  by  sections  parallel  to  the  base  :  required  the  height 
of  each  part. — Similar  solids  vary  as  their  cubes,  hence 
use  the  E  slide.     Over  20  of  D,  set  3  shares ;  under  2  is 
17.48 ;  under  1  is  13.86.     Subtracting  from  each  preced- 
ing, we  have  2.52  inches  height  of  lowest  part:    3.62 
second;  13.86  third. 

156.  A  person  has  a  solid  globe  of  wood,  7  inches  in 
diameter,  and  requires  another  twice  the  size :  required 
its  diameter.— Over  7  of  D,  set  1  of  E;  under  2  is  8.82 
inches. 

157.  Perceiving  a  chandelier,  suspended  from  a  church 
ceiling,  moving  slowly  backwards  and  forwards,  I  observed 
that  it  made  14  swings  per  minute :  what  was  the  height 
of  the  ceiling  from  the  floor,  supposing  the  centre  of  gra- 
vity of  the  chandelier  to  be  8  feet  from  the  pavement  ? — 
Ans.  68  feet. 

158.  A  person  lent  another  a  cubical  rick  of  hay,  mea- 
suring 10  feet  each  way,  which  he  repaid  with  3  others 


160  A  TREATISE   ON   A   BOX   Of 

of  the  same  shape :  what  was  the  measure  of  each  ? — • 
Ans.  6.93  feet. 

159.  Two  pipes,  each  2  inches  internal  diameter,  fill  a 
cistern  in  an  hour;  they  are  then  stopped,  and  five  smaller 
ones  are  opened  at  the  bottom  of  the  vessel,  which  they 
empty  in  the  same  space  of  time :  what  is  the  diameter 
of  these  smaller  pipes,  each  being  the  same  ? — Ans.  1.265 
inches. 

160.  The  arms  of  a  pair  of  scales  are  of  unequal  length ; 
a  quantity  of  sugar,  weighing  19  Ibs.  in  one  scale,  weighs 
only  16  Ibs.  in  the  other :  what  is  its  real  weight  ?   Take 
the  mean  proportional. — Ans.  17.43  Ibs. 

161.  There  is  a  glass  in  the  shape  of  a  frustum  of  a 
cone,  6  inches  deep ;  its  top  diameter  is  3  inches,  its  bot- 
tom 2 ;  if  I  pour  water  into  it  till  it  is  f  full,  what  will 
be  the  depth  of  the  liquor? — Ans.  5.12  inches. 

162.  Three  men  bought  a  tapering  piece  of  timber, 
which  was  the  frustum  of  a  square  pyramid :  each  side  of 
the  greater  end  was  3  feet,  of  the  less  1  foot,  the  length 
was  18  feet :  what  was  the  thickness  of  each  man's  piece, 
supposing  they  are  to  have  equal  shares  ? — Ans.  3.27, 
4.56,  and  10.17  inches. 

163.  The  sides  of  a  triangle  measure  6,  5,  and  3  feet; 
it  is  required  to  construct  another  that  shall  contain  3| 
times  as  much ;  determine  the  length  of  the  sides. — Ans. 
11.69,  9.75,  and  5.85. 

164.  The  mean  distance  of  Jupiter  from  the  sun  is  495 
millions  of  miles  :  how  many  years  is  this  splendid  lumi- 
nary in  traversing  his  orbit  ? — Ans.  11|  years. 

165.  How  many  gallons  will  be  contained  in  a  cyliu- 


INSTRUMENTS   AND   TUB    SLIDE-RULE.  161 

drical  vessel,  18  J  inches  diameter,  and  8 1  deep? — Ans. 
8  gallons. 

166.  A  globe  of  cast-iron  weighs  18  Ibs.  :  what  is  its 
diameter? — Ans.  5.09  inches. 

167.  What  is  the  diameter  of  a  silver  sphere,  weighing 
25  Ibs.  avoirdupois  ? — Ans.  5.02  inches. 

168.  Seven  men  bought  a  grindstone,  a  yard  in  dia- 
meter, for  a  guinea;  they  paid  3s.  each,  and  agreed  to 
grind  down  their  separate  portions  in  succession :    what 
was  the  diameter  of  the  stone  when  each  began  to  grind  ? 
—Ans.  36,  33.3,  30.4,  27.2,  23.57,   19.24,  and  13.6 
inches. 

169.  There  are  two  similar  cylinders;  the  length  of 
the  one  is  8  inches,  and  its  diameter  4;  the  other  is  2| 
times  the  size  :  determine  its  length  and  diameter. — Ans. 
Length  11.28,  diameter  5.64. 

170.  Two  spheres  of  brass  are  to  each  other  in  the  pro- 
portion of  5  to  7 ;  if  the  larger  measures  12  inches  round, 
what  is  the  circumference  of  the  smaller  ? — Ans.  10.72. 

171.  Five  men  bought  a  grindstone  16  inches  in  dia- 
meter, for  15s.    A  pays  Is.,  B  2s.,  C  3s.,  D  4s.,  and  E  5s. ; 
each  man  is  to  grind  down  his  portion  in  succession,  com- 
mencing with  A,  and  ending  with  E,  who  is  to  leave  4 
inches  unground  :  what  is  the  diameter  as  each  begins  to 
grind?— Ans.  16,  15.49,  14.42,  12.65,  9.8. 

172.  The  frustum  of  a  pentagonal  pyramid  measures 
10  inches  along  each  side  at  top,  and  15  at  bottom,  and 
the  depth  is  20  inches ;  if  I  put  into  it  a  solid  globe  of 
wrought  iron,  weighing  108  Ibs.,  and  then  pour  in  12 
gallons  of  water :  what  depth  of  the  vessel  will  remain 

unfilled  ? — Aus.  8.28  inches, 
u* 


162  A   TREATISE   ON    A  BOX    OF 

173.  A  globe  of  wood,  10  inches  diameter,  suspended 
in  a  lathe,  was  turned  down  into  smaller  globes,  by  4  men 
successively,  each  chipping  off  an  equal  portion :    what 
was  the  diameter  when  each  began,  supposing  the  last 
left  a  globe  of  2  inches  diameter  ?— Ans.  10,  9.09,  7.96, 
and  6.35. 

174.  If  into  a  soap-bubble  3|  inches  diameter,  I  blow 
^  of  a  pint  of  air,  what  will  then  be  its  circumference  ? — 
Ans.  14.49  inches. 

175.  One  evening  I  chanced  with  a  tinker  to  sit, 

Whose  tongue  ran  a  great  deal  too  fast  for  his  wit. 
He  talked  of  his  art  with  abundance  of  mettle, 
So  I  asked  him  to  make  me  a  flat-bottom'd  kettle. 
Let  the  top  and  the  bottom  diameters  be 
In  just  such  proportion  as  five  are  to  three. 
Twelve  inches  the  depth  I  proposed,  and  no  more; 
And  of  gallons  to  hold  seven-tenths  of  a  score. 
He  promised  to  do  it,  and  straight  to  work  went, 
Got  right  the  proportions,  but  wrong  the  content. 
He  alter' d  it  then,  and  the  quantity  found 
Correct,  but  the  top  measured  far  too  much  round ; 
Till,  making  it  either  too  big,  or  too  little, 
The  tinker,  at  last,  had  quite  spoil'd  his  fine  kettle. 
But  he  vows  he  will  bring  his  said  promise  to  pass, 
Or  else  that  he'll  waste  every  ounce  of  his  brass. 
So  to  save  him  from  ruin,  kind  friend,  find  him  out 
The  diameters'  length,  for  he'll  ne'er  do  it,  I  doubt. 
Ans.  15.06  bottom,  22.1  top. 


INSTRUMENTS   AND   THE    SLIDE-RULE.  163 


CASK  GAUGING. 

IT  has  been  stated,  that  casks  are  usually  gauged  by 
considering  them  under  four  varieties;  and  questions  121, 
122, 123, 124,  show  the  content  of  a  cask  of  given  dimen- 
sions under  these  four  varieties,  in  which  it  will  be  seen 
that  there  is  a  variation  of  10  gallons,  according  to  the 
different  form  under  which  the  cask  is  viewed.  By  con- 
sidering the  vessel  as  part  of  a  prolate  spheroid,  we  shall 
have  the  content  too  great ;  for  no  cask  is  so  much  curved 
towards  the  head  as  this  would  make  it.  The  middle 
frustum  of  a  parabolic  spindle  approaches  nearer  to  the 
shape  of  casks  in  general.  Two  frustums  of  a  paraboloid 
leave  too  sharp  a  ridge  in  the  middle ;  and  the  frustums 
of  two  cones  give  the  content  far  too  small,  and  would,  in 
themselves,  make  a  ridiculous  kind  of  barrel.  The  gene- 
rality of  casks  seem  to  be  a  compound  of  the  first  and 
fourth  varieties,  the  bung  part  being  spheroidal,  and  the 
extremities  conical.  If,  in  addition  to  the  bung  and  head, 
we  take  the  diameter  halfway  between  the  two,  then  the 
true  content  is  readily  found  by  the  general  rule  for  frus- 
tums ;  but,  as  in  practice,  except  with  open  casks,  it  is  a 
somewhat  tedious  process  to  obtain  this  middle  diameter 
perfectly  correct,  without  which  it  is  useless,  since,  by  the 
nature  of  the  formula,  the  content  is  made  to  depend 
upon  it  in  a.  fourfold  measur?;  and  as  the  determining  to 
which  of  the  varieties  any  given  cask  makes  the  nearest 
approach,  is  a  work  requiring  much  skill  and  judgment, 
various  writers  have,  from  time  to  time,  attempted  to  dis- 


164  A   TREATISE   ON   A  BOX   OP 

cover  a  rule  that  shall  be  an  approximation  for  all  casks. 
Dr.  Button's,  for  this  purpose,  is  represented  by  the  for- 
mula I  (39  B2  +  25  H2  -f  26  HB)  .000031473.  This, 
besides  being  very  laborious,  generally  gives  the  content 
too  small.  By  considering  the  bung  diameter  as  the  prin- 
cipal regulator,  and  by  combining  the  formulas  for  sphe- 
roidal and  conic  frustums,  we  shall  arrive  at  a  method 
•which  never  can  be  far  from  the  truth,  and  is  of  the 
easiest  application  possible,  as  it  may  be  put  under  the 
following  form,  with  a  whole  number  for  a  gauge  point,  a 
desideratum  in  all  cases  with  the  slide-rule. 

I  (H2  -f  2.B2) 

S32 

that  is,  over  33  on  D  set  the  length ;  then  the  number 
over  the  head  plus  twice  the  number  over  the  bung,  is 
the  content  in  gallons.  For  computation,  the  formula  is 
equally  simple  and  easy. 

I  (H2  -{-  2.B2)  .000919. 

EXAMPLE. 

176.  A  cask  measures  47  inches  long;  the  head  diame- 
ter is  26,  and  the  bung  31  inches :  required  the  content. 

Over  33  set  47 ;  then 

over  26  =  29.2 

31=41.5 

41.5 

112.2  gallons. 


INSTRUMENTS   AND   THE   SLIDE-RULE.  165 

Numerically,  26s  =  676 

31s  =961 

961 


2598  X  47  X  .000919  —  112.2 
gallons. 

This  is  an  example  of  a  cask  whose  middle  diameter 
was  found  to  be  29.3  as  nearly  as  possible,  the  content  of 
which  in  gallons,  by  formula  21,  will  be 

I  (Ha  -f  B9  -f  2n 


46a 
Over  46  place  47 ;  then 

over  26.    =15. 
31.    =21.4 
58.6=76.1 


112.5 

The  same  formula,  arranged  for  numerical  computa- 
tion, is 

Z(H'+B'+2^J').  0004721.* 

Example,  26»  =  676 
31»  =  961 
58.6' =3433.96 


5070.96  X47  X  .0004721  =  112.51 
gallons,  true  content. 


*  .00047^1  is  the  reciprocal  of  the  round  frustum  divisor  for 
gallons. 


166  A  TREATISE  ON   A  BOX   OF 

By  Dr.  Button's  Rule,  the  content  will  be — 
39  X  31a  =  37479 
25  X  26a  =  16900 
26X26X31  =20956 

75335  X47  X  .000031473  =  111.42 

gallons. 

The  following  Table  contains  50  casks  that  were  care- 
fully gauged  while  empty,  and  their  contents  subse- 
quently tested  by  actual  measurement  with  water.  A 
great  portion  are  taken  from  Dr.  Button's  works,  some 
from  Nesbit  and  Little's  gauging,  some  from  Todd's 
Manual,  and  the  rest  have  come  under  my  own  observa- 
tion. They  will  serve  as  exercises  for  the  student,  and 
show  the  value  and  efficiency  of  the  rule  I  have  proposed. 


INSTRUMENTS   AND   THE   SLIDE  RULE. 


Ib7 


TM      Cental. 
bj    R«l.    fcr 

f  ".•••.:-•        4 

Cntort  by  Dr 
H--I-  rt  M 

'.  r    5    ix« 

Coital  by  pw- 

rri  £z 

MM. 

No. 

L 

H 

B 

2m 

1 

28.3 

23.2 

27.7 

52.6 

54.41 

6351 

5351 

2 

29.8 

22^ 

26. 

49.6 

51.05 

60.41 

60.52 

3 

30.8 

23.2 

275 

52.2 

68.44 

57.78 

68.10 

4 

3i2 

24.5 

30.1 

56.8 

71.94 

70.51 

71^8 

5 

30. 

24.7 

29.2 

55.2 

63.87 

63.40 

63.80 

6 

32.5 

23.8 

28.2 

63.6 

64.97 

64.12 

6451 

7 

34.3 

26^ 

335 

fri2 

92.02 

00.72 

92^5 

8 

34.5 

26.4 

33. 

61.4 

90.49 

89.71 

91.01 

9 

41. 

26.3 

32.2 

60.4 

104.07 

102.41 

104.10 

10 

37. 

26.1 

31.8 

59.8 

92.03 

90.89 

91^5 

11 

44.5 

34.4 

40.8 

77.6 

186.34 

183.61 

184.53 

12 

47. 

26^ 

33.8 

62.8 

128.2 

WJH 

128.2 

13 

34.2 

27.2 

33.8 

6Z8 

MM 

93.21 

94.8 

14 

47. 

25.3 

32. 

69.4 

115.21 

113.92 

116. 

15 

45.5 

30.7 

38. 

71. 

159.54 

157.01 

159.56 

16 

44.6 

24.7 

323 

59.2 

108.47 

107.12 

109.82 

17 

48.6 

24.2 

32.1 

HJ 

116.40 

114.78 

117.61 

18 

46. 

25.7 

34.7 

63.4 

127.78 

125.11 

129.32 

19 

48.8 

24.2 

32.1 

58.8 

116.88 

115.28 

118.21 

20 

51.2 

23.3 

31. 

56.4 

113.24 

11253 

115.98* 

21 

48. 

BJ 

33.8 

62.8 

133.28 

132.03 

iiur* 

22 

51.6 

36.6 

41.6 

79.2 

22759 

228.52 

227.62 

23 

57. 

32.7 

42. 

78.2 

240.8 

235.43 

240.81 

24 

54. 

34.8 

44.8 

83. 

257.66 

253.41 

259^2 

25 

45.6 

28. 

34.6 

64.8 

133.04 

131.37 

133.05 

26 

45. 

27. 

36. 

66. 

13556 

133JZ5 

137.12 

27 

4'.: 

24.6 

30.9 

58. 

MJI 

106.00 

WJi 

28 

32.5 

21.4 

26.2 

49.6 

65.31 

63.87 

64.65 

29 

27.7 

19.6 

23.4 

44.2 

37.73 

37.33 

37.65 

30 

42. 

28. 

35. 

66.4 

124.64 

122.81 

124.62 

31 

50. 

26. 

32. 

60.2 

125.67 

123.16 

125.2 

32 

66. 

62. 

62. 

116.6 

627.63 

627.21 

630.31* 

33 

60. 

36. 

45. 

84, 

MSB 

290.11 

294.72 

34 

58.5 

4-  •.! 

49.8 

93. 

351.76 

349.42 

852.10 

35 

60. 

40. 

50. 

93.2 

362.17 

358.15 

363.75 

36 

44.5 

34.4 

40.8 

77.2 

185.04 

I8UU 

18454 

37 

355 

30. 

36. 

68. 

114.3 

112.97 

113.93 

38 

49.5 

23.3 

31.7 

58.6 

116.41 

112.18 

116J3 

39 

40. 

30. 

36. 

67.8 

UBJB 

127.32 

UKM 

40 

50. 

23. 

31. 

67.4 

112.94 

108.78 

112.62 

41 

54. 

39. 

45. 

86. 

-TV.'') 

276.31 

276.46 

42 

51. 

235 

31.5 

HJ 

119.3 

11551 

11851 

43 

39.6 

29.4 

355 

67. 

123.64 

121.92 

tMJg 

44 

39.5 

30. 

36.7 

69.2 

131.2 

ma 

130.46 

45 

40.5 

29.8 

35.5 

67. 

l2rV.> 

126.06 

126.86 

46 

41.5 

30. 

37. 

69.6 

130LM 

136.72 

138.75 

47 

51. 

23.7 

31.9 

59.4 

UBJfl 

117.73 

121.72 

48 

48. 

25.2 

33.4 

61.8 

126^2 

122.78 

126.4 

49 

56.4 

32.3 

41.2 

76.4 

Muaa 

225.21 

230.08 

50 

45.3 

28.2 

34.9 

66.2 

134.12 

13255 

134.52 

Total 

7415.12 

7312.05 

7423.04 

168  A   TREATISE  ON   A   BOX   OV 

In  47  out  of  the  50,  the  rule  I  have  proposed  agree? 
nearest  with  the  truth;  in  the  three  marked  with  an 
asterisk,  Dr.  Hutton's  comes  nearer.  The  total  error  by 
his  is  103  gallons ;  by  mine,  8  gallons.  In  setting  down 
the  dimensions,  the  most  concise  way  will  be  to  place  the 
length  on  the  left  hand,  with  a  brace  between  it  and  the 
diameters,  recollecting  that  33  is  the  gauge  point,  when 
three  dimensions  are  used;  and  46,  when  four  dimen- 
sions are  taken. 

EXAMPLE. 

177.  The  length  of  a  cask  is  40  inches,  the  head  30, 
the  bung  36,  and  twice  the  middle  67.8  inches :  required 
the  content. 

By  Proposed  Eule  for  3  dimensions. 
30=33.05 


40  {  36  =47.6 


47.7* 
128.35  gallons. 


By  General  Rule  for  4  dimensions. 

f30     =17. 
40^36    =24.5 
(67.8  =  86.9 


128.4  gallons. 

The  dimensions  of  casks  are  taken  most  readily  with 
the  long  and  cross  callipers,  and  the  bung  and  head  rods. 

*  If,  as  in  this  case,  the  number  found  standing  over  the  bung 
diameter  appear  to  be  more  than  47.6,  and  less  than  47.7,  set 
down  both,  as  above. 


INSTRUMENTS   AND   THE   SLIDE-RULE.  169 

When  these  are  not  at  hand,  their  place  may  be  supplied 
as  follows : — 

Procure  a  straight  piece  of  deal, 
about  |  of  an  inch  square,  and  6 
feet  long,  for  a  measuring-rod ; 
and,  with  a  camel's-hair  pencil  and 
Indian  ink,  divide  it  into  inches 
and  tenths.  Take  another  piece, 
AB,  an  inch  square,  and  about 
4  feet  long ;  and  near  one  end,  as 
at  a,  cut  a  notch,  and  2  inches  from  it,  b,  make  a  mark, 
and  place  a  cipher  0.  Then  divide  the  distance  from  b 
to  the  end  B  into  inches  and  tenths.  Also  procure  two 
pieces  of  string,  each  with  loops  at  one  end,  and  heavy 
plummets  of  lead  at  the  other.  Before  tying  the  loop,  ou 
one  of  the  strings  slip  3  pieces  of  cork,  e,  v,  -z,  about  i  of 
an  inch  thick,  and  }  of  an  inch  square.  Then, 

To  take  the  Dimensions  of  a  standing  Cask. 
With  a  piece  of  string  and  chalk,  by  problem  2,  page  20, 
strike  a  line  across  the  middle  of  the  head  of  the  cask ; 
lay  the  rod  AB  over  this  line,  and  bring  the  plummet 
depending  from  a  up  to  the  bulge  of  the  cask.  Then  slip 
the  other  plummet  along  to  c,  till  it  touches  the  cask  in  like 
manner.  The  number  now  cut  by  c  will  be  the  intemal 
bung  diameter  C,  the  distance  ab,  of  2  inches,  being  an 
allowance  for  twice  the  thickness  of  the  staves.  With 
the  measuring  rod  take  the  distance  from  y  (the  under  side 
of  the  rod  AB)  to  the  ground  f.  Also  the  distance  from  o 
(the  upper  side  of  the  rod)  to  n}  the  head  of  the  barrel. 
Then  yf  minus  twice  on,  will  be  the  internal  length  of  the 
cask,  the  thickness  of  the  square  rod,  AB,  being  supposed 

15 


170  A   TREATISE   ON  A   BOX   OF 

equal  to  the  thickness  of  the  head  of  the  cask,  which  is 
generally  1  inch.  To  take  the  middle  diameter  D,  slip 
the  top  cork  up  to  e,  till  the  distance  ye  is  equal  to  on. 
The  length  of  the  cask  being  known,  slip  the  second  cork 
down  to  v,  the  distance  ev  being  J  of  the  length;  in  the 
same  manner  adjust  the  cork  z,  if  deemed  necessary. 
Then  add  together  the  distances  vv,  zz,  and  subtract  their 
sum  from  the  bung  diameter,  or  deduct  twice  the  dis- 
tance fl.w,  if  the  curve  of  the  cask  be  uniform;  the 
remainder  will  be  the  middle  diameter,  D.  In  the  same 
way  might  a  diameter  be  taken  halfway  between  D  and  C. 
The  oblique  line  sx,  measured  from  the  inside  of  the 
chimb  to  the  outermost  sloped  edge  of  the  opposite  stave, 
will  be  the  internal  head  diameter ;  or  twice  the  distance 
at  e  may  be  deducted  from  the  bung.  If  only  three 
dimensions  are  taken,  the  corks  may  be  dispensed  with ; 
but  in  ullaging  standing  casks,  they  will  be  found  ex- 
tremely convenient. 

For  taking  the  dimensions  of  lying  casks,  a  common 
pair  of  callipers  may  be  made  by  any  carpenter,  as  annexed. 

kbc,  efq,  are  precisely  like 
a  carpenter's  square.  The 
arms  Ik,  ef,  may  be  an  inch 
.3  square,  and  2  feet  6  inches 
long ;  the  blades  be,  fq,  about  f  of  an  inch  thick,  and  an 
inch  broad.  At  c  and  q  two  pieces  are  fixed  at  right 
angles,  the  distance  cd  being  4  inches.  In  the  face  of  the 
arm  We,  let  a  groove  be  ploughed  and  worked  under  with 
a  side  tool,  to  a  dove-tailed  shape,  like  the  section  shown 
at  m.  The  under  side  of  the  arm  ef  is  to  be  cut  to  match 
it  like  the  section  shown  at  n.  The  arm  cf  will  now  slide 


INSTRUMENTS   AND   THE   SLIDE-RULE.  171 

along  the  arm  Nc ;  in  fact,  it  would  be  preferable  if  it  were 
cut  like  a  slide-rule,  but  carpenters  have  not  tools  for 
effecting  this.  One  inch  from  a  (which  is  opposite  to  d) 
make  a  mark,  and  place  a  cipher  0.  Then  from  0  to  k 
will  be  25  inches ;  divide  this  into  inches  and  tenths,  and 
number  it  from  0  towards  k.  On  the  arm  ef  at  the  point 
opposite  to  h  make  a  mark,  and  1  inch  from  it  toward  e 
place  25  ;  then  divide  the  space  from  25  to  e  into  inches 
and  tenths,  and  number  them  backward.  When  this  arm 
is  made  to  slide  in  the  other,  and  drawn  out  to  measure 
the  length  or  bung  diameter,  the  number  standing  oppo- 
site the  end  k  will  denote  such  length  or  bung  diameter. 

To  find  the  content  of  a  large  circular  vessel,  that  ap- 
pears to  bulge  irregularly,  by  an  odd  number  of  equidis- 
tant diameters. 

178.  Let  the  vessel  be  the  cask  on  page  169,  and  let 
there  be  taken  9  diameters,  commencing  with  the  head, 
level  with  <?,  and  proceeding  with  one  between  that  and  D, 
down  to  the  bottom,  which  suppose  to  be  80,  83,  86,  88, 
90,  89,  87,  84,  and  81  inches,  and  the  depth  of  the  vessel 
96  inches,  consequently,  the  common  distance  of  the  dia- 
meters 12  inches. 

Place  in  a  line  the  letters       x?  4e"  2o* 

Under  x3  place  the  square  of  the  first  or  top  diameter; 
under  4e9  the  square  of  the  second  diameter;  under  2o* 
the  square  of  the  third  diameter ;  under  4^  the  square  of 
the  fourth  diameter ;  and  so  on,  alternately,  to  the  last, 
the  square  of  which  place  under  x9,  along  with  the  other 
extreme.  -  Add  together  the  three  columns  separately, 
and  multiply  that  under  4<r  by  4 ;  and  that  under  2o* 


172  A  TREATISE   ON   A   BOX  OF 

by  2.     Add  the  three  together,  multiply  by  the  common 
interval,  and  divide  by  the  cone  divisor 


EXAMPLE. 

20s 


6400  6880  7396 

6561  7744  8100 

7921  7569 


12961  7065 


23065 

29610  2 

4  

46130 


] 18440  

46130 
12961 

177531  Xl2-=-1059.108  =  2011}  gallons. 

This,  it  will  be  seen,  is  merely  a  modification  of  the 
general  rule  for  frustums.  For,  let  the  diameters,  taken 
in  order,  be  a,  b,  c,  d,  e,  &c.  Then,  taking  three  at  a 
time,  we  have  a2  +  4Z>3  -f  c3;  c9  -f  4da  -f-  e3;  e3  -f  4/» 
-f-  ga,  &c.  j  that  is,  a8  -f-  4Z*9  -f  2d>  -f-  4d*  +  2^  +  4/a 
-\-ga',  namely,  the  square  of  the  extremes,  plus  4  times 
the  square  of  the  even  diameters,  plus  twice  the  square  of 
the  remaining  odd  diameters.  By  the  slide-rule  the  con- 
tent may  be  found  by  taking  it  as  three  successive  frus- 
tums. The  same  rule  obviously  applies  to  the  ullaging 
of  a  standing  cask. 

EXAMPLE. 

179.  The  depth  of  liquor,  in  a  cask  partly  filled,  is  20 
inches ;  five  oquidistant  diameters,  measured  from  the  sur- 


face  downward,  are  28,  27,  26,  24,  and  22  inches :  rt 
quired  the  content. 

x*  4e«  2o* 

784               ^729                   676 
484  576  2 

1268      1305       1352 
4 

"5220 
1268 

1352 

7840  X  5  -~-  1059.108  =  37  gallons. 

For  ullaging  a  lying  cask,  the  following  role  may  be 
employed. 

From  10  times  the  wet  inches,  subtract  the  bnng ;  mul- 
tiply the  remainder  by  the  content,  and  divide  by  8  times 
the  bang;  the  quotient  gives  the  liquor  in  the  cask ;  i.  e. 

(IQW-B)C 
SB 

To  find  the  content  of  vessels  whose  bases  are  nearly  of 
an  elliptical  form,  proceed  for  the  area  of  the  base  as  di- 
rected on  pa^e  1*29,  and  (after  multiplying  by  the  com- 
mon distance  of  the  ordinates,)  instead  of  dividing  by  3, 
multiply  by  the  depth  of  the  vessel,  and  divide  by  the 
pyramid  divisors,  these  being  equal  to  3  times  the  prism 
divisors.  If  the  vessel  also  bulge  up  the  sides,  take  an 
odd  number  of  equidistant  areas,  and  proceed  as  in  the  last 
example.  And  thus  may  any  solid  be  measured  ;  always 
observing,  that  when  equidistant  area*  are  taken,  the^ro- 
»i/</  divisors  must  be  employed ;  and  when  the  squares  of 


174  A  TREATISE  ON  A  BOX  OF 

equidistant  diameter*  are  used,  the  cone  divisors  must  be 
selected,  for  the  obvious  reason  that  thrice  the  latter  re- 
duce squares  to  circles. 


TIMBER  MEASURE. 

To  find  the  superficial  content  of  a  plank. 

Take  the  length  in  feet,  and  the  breadth  in  inches; 
then  divide  (by  12)  the  product  of  the  dimensions.  If 
the  board  tapers  regularly,  take  half  the  sum  of  the  end 
breadths  for  the  mean  breadth. 

EXAMPLE. 

180.  A  plank  16  feet  6  inches  long,  is  10  inches  broad 
at  one  end,  and  18  at  the  other :  what  is  the  content  ? — 
Here  14  is  the  mean  breadth.  Then  to  12  of  A  set  16*, 
and  under  14  is  19 \  square  feet.* 

*  It  is  much  to  be  regretted  that  the  foot  is  not  divided  into 
100  equal  parts  instead  of  96,  as  at  present.  The  mode  of  work- 
ing duodecimals,  though  simple  enough  in  itself,  often  leads  to 
confusion,  from  the  singular  names  given  to  the  result.  Thus, 
a  piece  of  wood  measures  9  feet  5  inches  by  3  feet  8  inches,  which 
multiplied  together,  according  to  the  prescribed  rules,  gives 
what  are  called  34  feet  6  inches  and  4  par  it.  Now,  these  inches, 
as  they  are  termed,  are  merely  twelfths  of  a  superficial  foot ;  and 
these  parts,  twelfths  of  such  twelfths ;  that  is,  duodecimal  frac- 
tions, each  number  decreasing  in  a  twelvefold  proportion  from 
left  to  right,  as  decimal  fractions  decrease  in  a  tenfold  propor- 
tion. The  name  of  inches,  given  to  the  6,  conveys  no  kind  of 
idea ;  for  they  are  neither  inches  nor  feet,  but  a  mixture  of  the 


INSTRUMENTS   AND   THE   SLIDE-RULE.  J15 

To  find  the  content  of  hewn  or  four- sided  timber. 

Take  the  length  in  feet,  the  breadth  and  thickness  in 
inches.  Find  a  mean  proportional  between  the  breadth 
and  thickness ;  then  divide  (by  12*)  the  length  multiplied 
by  the  square  of  the  mean  proportional.  If  the  tree 
tapers  regularly  from  end  to  end,  find  the  mean  pro- 
portional between  the  mean  breadth  and  thickness. 

EXAMPLE. 

181.  A  log  of  wood  is  23  feet  6  inches  long,  15  inches 
thick,  and  22  broad  :  required  the  content. — Over  15  of 
D  set  15,  and  under  22  is  18.17,  a  mean ;  then  over  12  of 
D  set  23  J,  and  over  18.17  is  53|  solid  feet. 

182.  The  length  of  a  piece  of  timber  is  23.8  feet;  the 
breadth  at  the  greater  end  is  20.18  inches,  at  the  less 
16.42  inches;  the  thickness  at  the  greater  end  is  14.12 
inches,  at  the  less  10.48  inches :  required  the  content. — 
Here  20.18  -j-  16.42  =  36.6,  the  half  of  which  =  18.3  tho 

two,  the  content  being,  in  reality,  34  square  feet,  together  with 
another  piece,  1  foot  long  and  6  inches  broad ;  and  another,  1 
foot  long,  and  ^  of  an  inch  broad ;  a  square  foot,  in  fact,  being 
the  integer,  and  the  others  successive  twelfths.  For  the  car- 
penter the  present  nomenclature  answers  well  enough,  as  he 
perfectly  understands  that  it  is  a  trifle  more  than  34  feet  and  a 
half,  which  is  sufficient  for  his  purpose.  But  a  misconception 
of  the  principle  of  duodecimals,  carried  out  by  unskilful  people, 
has  led  to  the  wildest  confusion;  for  even  in  some  works  on 
arithmetic,  designed  for  the  instruction  of  the  young,  occurs  the 
unimaginable  problem  of  multiplying  half-a-crown  by  half-a- 
crown  ;  the  result  of  which  notable  achievement  is  stated  to  b« 


176  A   TREATISE   ON   A  BOX  OF 

mean  breadth ;  also,  14.12  -f  10.48  =  24.6,  the  half  of 
which  =  12.3.  Over  12.3  of  D  set  12.3,  and  under  18.3 
is  15,  a  mean  proportional :  then,  over  12  of  D  set  23.8, 
and  over  15  is  37.2. 

To  find  the  content  of  round  timber.  Take  the  length 
in  feet,  and  the  girt  in  inches :  then  divide  (by  12s)  the 
length  multiplied  by  the  square  of  the  quarter  girt. 


EXAMPLE. 

183.  Required  the  content  of  a  tree  48  feet  long,  the 
girts  at  the  ends  being  60  and  18  inches. — Here  39  is  the 
mean  girt,  t  of  which  =  9. 7 5.     Then,  over  12  of  D  set 
48,  and  over  9.75  is  31.7  feet  nearly. 

The  above  rule  gives  only  about  ^  of  the  true  content, 
but  is  adopted  in  practice,  as  it  compensates  the  purchaser 
for  the  waste  of  timber  occasioned  by  squaring  it.  The 
following  rule  gives  the  true  content  very  nearly.  Divide 
(by  12a)  twice  the  length,  multiplied  by  the  square  of  } 
of  the  girt. 

EXAMPLE. 

184.  Required  the  content  of  the  last-mentioned  tree. — 
Here  }  of  39  =  7.8.    Hence,  over  12  of  D  set  96  (twice 
the  length)  and  over  7.8  is  40.56  cubic  feet. 

But  neither  this,  nor  the  rule  for  squared  timber,  is 
quite  correct,  if  the  tree  tapers,  but  is  sufficiently  so  for 
all  practical  purposes. 


INSTRUMENTS    AXD   THE    SLIDE-RULE. 


177 


LAND  SURVEYING. 


Dm  =  180 
Bn  =208 


c 

543 

A 


A 
244 
D 


D 

422 
C 


C 

300 
136 

H 

B 


•J  > 
93 

A 


Set  up  poles  at  A,  B,  C,  and  D,  so  that,  standing  at  A, 
you  can  see  B  and  D,  the  end  of  the  sides  whose  meeting 
forms  the  angle :  and  so  of  the  others.  And  suppose  the 
hedge  to  run  on  straight,  or  nearly  so,  from  A  to  a,  then  to 
bend  and  run  on  straight  to  c,  and  so  on.  Form  a  field- 
book,  as  on  the  right  of  the  diagram,  by  ruling  two  lines 
down  the  middle  of  a  page;  and,  in  using  this,  begin  at 
the  bottom  and  write  upward,  placing  the  main  lines  in 
the  middle,  and  the  offsets  right  or  left,  as  they  are  on  the 
right  or  left  of  the  line  measured.  Then,  suppose  you 
commence  surveying  at  A ;  let  your  attendant  lead  the 
chain  toward  B ;  and  when  he  gets  it  extended,  see  that 
he  is  in  a  straight  line  between  yourself  and  B,  directing 


178  A  TREATISE   ON   A  BOX   OF 

him  by  a  wave  of  the  hand,  right  or  left,  according  as  you 
wish  him  to  move  to  one  side,  or  the  other.  His  position 
being  correct,  he  is  to  place  an  arrow  in  the  ground,  and 
walk  on  till  the  chain  is  again  extended,  when  he  places 
another  arrow,  while  you  take  up  the  first;  and  so  proceed. 
But  when  you  arrive  at  5,  opposite  to  a,  measure  ab,  (at 
right  angles  to  AB,)  with  an  offset  staff,  which  may  be  a 
thin  piece  of  deal  10  links  long.  Now,  suppose  from  A 
to  b,  92  links,  and  ai32  links;  place  the  92  in  the  middle 
column  over  A,  and  32  on  the  left  hand  of  it ;  and  so  pro- 
ceed till  you  arrive  at  the  end  of  B,  which  suppose  384 ; 
set  this  down  in  the  field-book,  and  over  it  place  the  letter 
B,  and  above  this  draw  a  line  across  the  page.  The  A 
being  placed  at  the  bottom,  and  B  at  the  top,  shows  that 
the  intervening  numbers  are  the  measures  of  the  AB  line. 
Proceed  with  the  rest  in  like  manner,  making  the  circuit 
of  the  field,  and  returning  to  A.  Then  from  A,  measure 
the  diagonal  AC.  With  these  dimensions  plot  the  field 
from  a  scale  of  equal  parts,  (feather-edged  plotting  scales 
are  best  for  this  purpose,)  and  drop  the  perpendiculars 
Dm,  B»,  and  from  the  scale  ascertain  their  lengths.  These 
are  set  down  underneath  the  diagram,  as  they  are  not  sup- 
posed to  have  been  .measured  in  the  field;  but  if  a  cross 
staff,  or  theodolite,  be  employed,  they  are  to  be  taken 
while  proceeding  along  the  diagonal,  and  set  down  in  the 
field-book,  like  offsets,-  and  then  the  sides  AD,  DC  will 
not  require  measuring,  supposing  there  are  no  offsets  on 
them. — To  find  the  area.  Add  together  Dm,  B?*,  and 
multiply  their  sum  by  AC.  For  the  offsets ;  for  the  tri- 
angle A  ab,  multiply  Ab  by  ba.  For  the  succeeding 
trapezoid,  add  together  ab,  cd,  and  multiply  by  bd ;  and 
go  proceed.  Then,  as,  in  all  these  cases,  this  gives  double 


INSTRUMENTS   AND   THE   SLIDE-RULE.  179 

the  area,  add  the  whole  together,  halve  the  sum,  and  di- 
vide by  the  number  of  links  in  an  acre,  viz.  100,000,  (that 
is,  point  off  the  5  right-hand  figures,)  and  the  content  is 
in  acres  and  decimal  parts;  the  latter  of  which  being 
multiplied  by  4  and  40,  (which  need  not  be  set  down,) 
gives  the  roods  and  poles.  In  taking  the  distances  from 
the  field-book,  the  numbers  up  the  middle  column,  92, 
208,  &c.,  have  to  be  subtracted  from  each  succeeding,  and 
the  offsets,  0,  32,  36,  &c.,  to  be  added  together  in  pairs. 
The  work  will  stand  thus  : — 


180 

92 

116 

176 

136 

210684 

208 

32 

68 

36 

33 

2944 











7888 

388 

194 

928 

1056 

408 

6336 

543 

276 

696 

528 

408 

4488 

1164 

2944 

7888 

.6336 

4488 

2)232340 

1652 

1940 

1.16170 

210684 

25.8720 

Content,  1  acre  25  poles. 


180  A  TREATISE   ON    A   BOX   Off 


TRIGONOMETRY  AND  NAVIGATION. 


THE  trigonometrical  slide  is  a  slide  containing  the  loga- 
rithmic sines  and  tangents,  the  former  of  which  work  to 
the  line  D,  and  the  latter  to  the  line  A,  which  lines,  as 
before  explained,  are  also  logarithmic.  But  it  is  to  be 
recollected,  that  it  is  the  distances  only  that  are  loga- 
rithmic, not  the  numbers ;  hence,  when  the  slide  is  laid 
evenly  in,  then  the  numbers  on  A  are  the  natural  tan- 
gents, and  the  numbers  on  D  the  natural  sines  of  the 
degrees  marked  on  the  slide  :  when  in  any  other  position 
they  are  proportional  to  the  natural  sines  and  tangents  of 
those  degrees;  and,  therefore,  if  we  set  the  first  term  of 
a  proportion  over  or  under  the  second,  then  the  third  will 
stand  over  or  under  the  fourth,  the  fifth  over  or  under  the 
sixth,  and  so  on.  In  making  use  of  the  tangent  line, 
three  points  may  be  taken  as  radius,  either  the  beginning, 
middle,  or  end  of  the  slide ;  but  the  middle  point, 
marked  45°,  will  in  practice  be  most  convenient.  In 
using  the  sine  slide,  two  points  may  be  taken  as  radius, 
either  the  beginning  or  the  end  of  the  slide,  as  may  be 
found  necessary  for  preventing  the  numbers  from  over- 
running. Having  already  given  several  questions  under 
the  sector,  and  Navigation  being  only  an  application  of 
Trigonometry,  it  will  be  sufficient  here  to  shew  the  mode 


INSTRUMENTS   AND   THE   SLIDE-RULE.  181 

of  working  an  example  or  two  with  the  slide-rule,  which 
the  student  will  find  infinitely  superior  for  the  purpose. 
Take  the  tower  at  page  68.  By  the  sine  line,  sin.  42  J  : 
200  :  :  sin.  47  J  :  height, 

sin.  42  J       sin.  47  \      .,      , 
or     onn     =  .    .  ,     •    therefore — 
200  height  ; 

Under  42 1  of  the  sines  bring  200;  then  under  471  ia 
218.2,  the  height. 

By  the  tangent  line,  making  AC  radius — 

Rad.  or  tan.  45°  :  200  : :  tan.  47 J  :  height; 

200  height 

tan.  45       tan.  47  J" 

Over  45  of  the  tangents  set  200 ;  then  over  47  J  is  218.2, 
the  height,  (as  before.) 


182 


A   TREATISE   ON   A   BOX    O? 

NAVIGATION. 


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INSTRUMENTS   AND  THE   SLIDE-RULE.  1P3 

PLANE  SAILING. 

Plane  sailing  supposes  the  earth  to  be  a  plane,  the  meri- 
dians parallel  to  each  other,  and  the  lengths  of  degrees 
everywhere  equal ;  and  involves  the  consideration  of  four 
quantities,  difference  of  latitude,  nautical  distance,  de- 
parture, and  course.  Let  K  (diagram  p.  48,  disregarding 
the  circle,)  denote  a  point  on  the  earth's  surface,  and  KG 
its  meridian.  Draw  a  line  from  K  to  A,  and  suppose  a 
ship  to  sail  along  it  from  K  till  she  arrive  at  A ;  then  KA 
will  be  the  distance  sailed ;  DA,  the  departure  from  the 
meridian ;  KD,  the  difference  of  latitude ;  and  the  angle 
DKA,  contained  between  the  meridian  and  the  rhumb 
sailed  on,  the  course.  The  difference  of  latitude  is  thus 
represented  by  a  vertical  line,  the  departure  by  a  horizontal 
one,  the  distance  by  the  hypothenusal  line  forming  with 
the  other  two  a  right-angled  triangle,  and  the  course  by 
the  angle  included  between  the  difference  of  latitude  and 
the  distance.  Then,  if  we  make  distance  radius,  the  de- 
parture becomes  the  sine,  and  the  difference  of  latitude  the 
cosine,  of  the  course ;  or,  if  diff.  lat.  be  made  radius,  de- 
parture becomes  the  tangent  of  the  course. 

EXAMPLES. 

1.  A  ship  sails  38°  S.,  255  miles  W. :  required  the  diff 
of  lat.  and  departure.    Complement  of  38°  =  52°,  then — 

Sin.  90  :  255  : :  sin.  52°  :  diff.  lat.  : :  sin.  38°  :  dep 
Under  90  of  the  sines  set  255 ;  then — 

Under  52°  is  201  miles,  diff.  lat. 
and  under  38°  is  157  miles,  departure. 


Ib4  A  TREATISE   ON    A   BOX   OF 

2.  A  ship  sails  from  lat.  44°  50'  N.  between  S.  and  E. 
till  she  has  made  64  miles  of  easting,  and  is  then  found 
to  be  in  lat.  42°  56'  N.  :  required  the  course  and  distance. 

44°  50' 
42    56 


1    54  =  114  miles,  diff.  lat. 


As  114  :  rad.  : :  64  :  tan.  of  course. 
Under  114  set  45°  of  the  tangents,  then  under  64  is 
29°  20',  the  course. 

Again,  sin.  29°  20' :  64  : :  sin.  90°  :  dist. 

Under  29°  20'  set  64,  then  under  90°  is  130.6  miles, 
distance. 

3.  A  ship  in  lat.  45°  25'  N.  sails  N.E.b.N.  1  E.  till  she 
comes  to  46°  55'  N. :  required  the  distance  and  departure. 
N.E.b.N.  J  E.  =  39°  22  J',  comp.  of  which  =  50°  371'. 

4  6°  55' 
45  25 


1    30  =  90  miles,  diff.  lat. 


Sin.  &0°  371'  :  90  miles  : :  sin.  39°  221'  :  73.8  miles  de- 
parture : :  sin.  90°  :  116.4,  distance.* 

*  As  the  learner  is  supposed  by  this  time  to  be  familiar  with 
the  mode  of  operation,  it  will  be  sufficient  for  the  future  to  indi- 
cate the  proportion,  without  repeating  the  directions  for  setting 
the  slide.  Thus,  in  the  above  instance,  under  50°  37£'  set  90 
miles,  then  under  39°  22J'  will  be  73.8,  and  under  90°  will  be 
1 J  6  4  miles ;  and  so  of  all  others.  When  the  word  rad.  occurs 
as  the  first  or  second  term,  before  adjusting  the  slide  run  the 
eye  along  the  proportion  to  see  if  the  word  sin.  or  tan.  follows, 


INSTRUMENTS   AND  THE   SLIDE-RULE.  185 

EXAMPLES   FOR   PRACTICE. 

4.  A  ship  sails  from  lat.  56°  50'  N.  on  a  rhumb  be- 
tween S.  and  S.  W.  126  miles,  and  is  then  found  to  be  in 
lat.  55°  40' :  required  the  course  she  sailed,  and  her  de- 
parture   from    the    meridian. — Ans.    Course,    56°  15'; 
departure,  104.8  miles. 

5.  A  ship  in  lat.  44°  50'  N.  sails  S.  29°  20'  E.  130.8 
miles :  required  diff.  lat.  and  departure. — Ans.  64  dep.  : 
114  diff.  lat. 

6.  A  ship  in  lat.  45°  25'  N.  sails  N.E.b.N.  *  E.  116.4 
miles :  required  dep.,  diff.  lat.,  and  latitude  come  to. — 
Ans.  74  dep. ;  90  miles,  or  1°  30'  diff.  lat ;  and  46°  55'  N. 
lat.  come  to. 

7.  A  ship  at  sea  sails  from  lat.  34°  24'  N.  between  N. 
and  W.  124  miles,  and  is  found  to  have  made  86  miles 
of  westing  :  required  the  course  steered,  and  diff.  of  lat., 
or  northing  made  good. — Ans.  Course,  43°  54';  diff.  lat. 
1°29';  35°  53' N.  lat.  come  to. 

8.  A  ship  in  lat.  24°  30'  S.  sails  S.E.b.S.  till  she  has 
made  96  miles  of  easting :  required  the  distance  sailed, 
and  diff.  of  lat.  made  good. — Ans.  Diff.  lat.  143.7;  dis- 
tance, 172.8 ;  lat.  come  to,  26°  54'  S. 


and  use  the  sine  or  tangent  line  accordingly.  And  in  every 
case  it  will  be  advisable  for  the  beginner  to  construct  a  diagram, 
as  nothing  tends  so  much  to  make  the  operation  perfectly  un- 
derstood ;  and  what  is  thoroughly  understood  at  the  commence- 
ment is  seldom  afterwards  forgotten. 

16* 


186 


A  TREATISE   ON   A  BOX   OF 


TRAVERSE  SAILING. 

When  a  ship  sails  upon  several  courses,  the  zigzag  lina 
fche  describes  is  called  a  traverse ;  and  the  reducing  the 
courses  into  one,  and  thereby  finding  the  course  and  dis- 
tance made  good  upon  the  whole,  is  called  the  resolving 
of  the  traverse.  For  this  purpose,  construct  a  table  of  six 
columns,  in  the  first  of  which  is  the  course,  and  in  the 
second  the  distance ;  then  find  the  diff.  lat.  and  dep.  for 
each  course,  and  enter  it  N.  or  S.,  E.  or  W.,  as  it  may  be. 
Add  up  the  columns  separately ;  the  difference  of  the  third 
and  fourth  will  give  the  diff.  of  lat.,  and  the  diff.  of  the 
fifth  and  sixth,  the  departure.  Then,  having  obtained 
the  total  diff.  lat.  and  dep.  which  the  ship  has  made,  find 
the  corresponding  course  and  distance. 

EXAMPLE. 

9.  A  ship  from  the  equator  sails  N.  48,  W.  37,  N.W.  18, 
N.E.  70,  N.N.E.  24,  and  E.  32  miles:  required  her 
course,  distance,  and  latitude  reached. 


Course. 

Dist. 

Diff.  Lat. 

Dep. 

N. 

S. 

E. 

W. 

N. 
W. 
N.W. 

N.E. 
NN.E. 
E. 

48 
37 
18 
70    , 
24 
32 

48 

12.72 
49.5 
22.18 

— 

49.5 
9.18 
32. 

37 
13.72 

132.4 

90.68 
49.72 

49.72 

40.96 

INSTRUMENTS   AND   THE   SLIDE-RULE.  187 

First  she  sails  due  N.,  and  so  will  have  no  departure : 
therefore  place  48  under  N. 

Her  second  course  is  due  W.,  and  so  she  will  have  no 
diff.  of  lat. ;  therefore  place  37  under  W. 

Her  third  course  is  45°,  and  therefore  her  departure 
and  diff.  of  lat.  will  be  equal.  Over  18  place  90°  of  the 
sines,  and  under  45°  is  12.72,  which  place  under  N.  and  W. 

Her  fourth  course  is  also  45°,  and  therefore  her  dep. 
and  diff.  of  lat.  will  be  equal.  Over  70  place  90°,  and 
under  45°  is  49.5,  which  place  under  N.  and  E. 

Her  fifth  course  is  22  J,  cos.  of  which  =  sin.  67 J. 
Over  24  place  90°,  then  under  22  J  is  9.18,  her  departure, 
which  place  under  E. ;  and  under  67  J  is  22.18,  her  diff. 
lat.,  which  place  under  N. 

Her  last  course  is  32  due  E.,  and  so  she  will  have  no 

• 

diff.  of  lat. :  therefore  place  32  under  E. 

Add  up  the  three  columns.  As  there  is  no  number 
standing  under  S.  the  diff.  of  lat.  is  132.4  =  2°  12'  N. 

Subtract  the  W.  from  the  E.,  and  the  remainder  is 
40.96  E.  for  the  total  departure  :  then — 

As  132.4  :  rad.  : :  40.96  :  tan.  17°  12',  the  course. 

Again,  sin.  17°  12f  :  40.96  :  :  sin.  90°  :  138.6  wiles, 
the  distance. 

EXAMPLES   FOR  PRACTICE. 

10.  A  ship  from  Cape  Clear,  lat.  51°  25'  N.,  sails 
SS.E.  J  E.  16,  E.S.E.  23,  S.W.b.W.J  W.  36,  W.t  N.  12, 
and  S.E.b.E.  JE.  41  miles :  required  the  equivalent  course 
and  distance,  and  the  latitude  of  the  place  which  the  ship 
has  arrived  at. — Ans.  Course,  18°  12' ;  distance  62.75 
miles ;  lat.  in,  50°  25'  N. 


188  A   TREATISE   ON   A  BOX    OF 

11.  From  Cape  St.  Vincent,  in  lat.  37°  2'  N.,  a  ship 
sailed  S.W.b.S.  49,  S.b.E.  56,  S.E.b.E.  38,  S.W.  84, 
NN.W.  72,  and  E.N.E  24  miles :  required  the  course, 
distance,  and  latitude  come  to. — Ans.  Course,  26°  15' ; 
distance,  112  ;  lat.  in,  35°  22'  N. 


PARALLEL  SAILING. 

Since  the  meridians  meet  at  the  poles,  it  follows  that 
the  length  of  a  degree  on  any  parallel  of  latitude  dimi- 
nishes as  it  recedes  from  the  equator.  To  ascertain  this 
diminution,  when  a  vessel  sails  on  a  parallel  of  latitude, 
or  changes  her  longitude  only,  is  the  object  of  parallel 
sailing.  Let  FA  (diagram,  page  48,)  represent  the  earth's 
semi-axis ;  FCB,  a  quadrant  of  a  meridian;  B,  a  point  on 
the  equator;  C,  a  point  on  the  meridian,  and  conse- 
quently the  arc  CB,  or  angle  CAB,  the  latitude  of  C ; 
and  let  the  quadrant  revolve  on  AF;  then  the  circles 
described  by  the  points  C,  B,  or  similar  parts  of  them, 
will  be  proportional  to  their  radii  EC,  AB. 

Now  AB,  or  AC  :  EC,  or  AD  : :  rad.  :  cos.  CAB ; 
that  is,  difference  of  longitude,  or  distance  between  any 
two  meridians  on  the  equator,  or  parallel  described  by  B 
:  the  distance  between  those  meridians  on  the  parallel 
described  by  C  :  :  radius  :  the  cosine  of  the  latitude ;  or 
the  lengths  of  degrees  on  different  parallels  vary  as  the 
cosines  of  the  latitudes.  Hence,  if  in  any  right-angled 
triangle  ADC,  the  acute  angle  at  the  base  CAD,  be  made 
equal  to  the  latitude,  and  the  length  of  the  base  AD  equal 
to  the  departure,  or  meridian  distance,  or  distance  be- 
tween any  two  meridians  on  a  parallel  of  that  latitude ; 


INSTRUMENTS  AXD   THE    SLIDE-RULE.  189 

then  the  hypothenuse  AC  will  be  equal  to  the  arc  of  the 
equator,  or  the  difference  of  longitude  corresponding  to 
that  meridian  distance. 

EXAMPLES. 

12.  Required  the  number  of  miles  contained  in  a  degree 
of  longitude,  in  lat.-  55  N. 

cos.  55°  =  sin.  35°. 
sin.  90°  :  60  miles  : :  sin.  35°  :  34.4  miles. 

13.  A  ship  from  lat.  42°  52' N.  in  long.  9°  17' W. 
sails  due  W.  342  miles :  required  the  longitude  come  to. 

cos.  42°  52'  =  sin.  4 7°  8'. 
sin.  47°  8'  :  342  : :  sin.  90°  :  467,  diff.  long. 

467  =  7°  47' 
9   17 


17     4  W.  long,  come  to. 

14.  A  ship  sailed  224  miles  upon  a  due  "W.  course,  and 
by  observation  found  she  had  differed  her  longitude  6°  18', 
or  378  miles :  required  latitude. 

378  :  sin.  90°  : :  224  :  sin.  36°20/; 
and  sin.  36°  20'  =  cos.  53°  40',  the  latitude  required. 

15.  Two  ships  in  lat.  46°  30'  N.,  distant  asunder  654 
miles,  sail  both  directly  N.  256  miles :  required  their 
distance. 

256  =    4°  16' 
46    30 


50    46  N.,  lat.  reached. 


190  A  TREATISE   ON   A   BOX   OP 

Then  cos.  46°  30'  :  654  miles  :  :  cos.  50°  46';  or 
sin.  43°  30'  :  654  :  :  sin.  39°  14'  :  601  miles,  the  dis- 
tance. 

16.  Two  ships  in  lat.  45°  44'  N.,  distant  846  miles, 
sail  directly  N.  till  the  distance  between  them  is  624 
miles  :  required  the  lat.  reached  and  dist.  sailed. 

Cos.  45°  44'  =  sin.  44°  16';  then  846  :  sin.  44°  16'  :  : 
624  :  sin.  31°;  and  sin.  31°  =  cos.  39°,  lat.  come  to. 

Then  59°  0' 
45  44 


13    16  =  796  miles,  dist.  sailed. 


EXAMPLES   FOR  PRACTICE. 

17.  A  ship  in  lat.  54°  20'  N.  sails  directly  W.  on  that 
parallel  till  she  has  differed  her  longitude  12°  45' :  re- 
quired the  distance  sailed. — Ans.  446  miles. 

18.  A  ship  from  Cape  Finisterre,  lat.  42°  52'  N.,  long. 
9°  17'  W.,  sailed  due  "W.  342  miles :  required  the  longi- 
tude come  to. — Ans.  17°  4'  W. 

19.  A  ship  sails  on  a  certain  parallel  directly  W.  624 
miles,  and  has  then  differed  her  longitude  18°  46',  or  1126 
miles :  required  the  latitude  of  the  parallel  sailed  on. — 
Ans.  56°  20'. 

20.  A  ship  from  a  port  in  lat.  54°  N.  sailed  due  E.  200 
miles ;  then,  having  run  due  S.  an  unknown  number  of 
miles,  sailed  W.  250  miles,  and,  by  observation,  found 
she  had  arrived  at  the  meridian  of  the  port  she  sailed 
from  :  required  the  lat.  come  to,  and  distance  run  in  tho 
S.  direction. — Ans.  42°  43'  lat.  come  to;  677  miles  run 


INSTRUMENTS  AND   THE   SLIDE-RULE.  191 


MIDDLE  LATITUDE  SAILING. 

Middle  Latitude  Sailing  is  a  composition  of  plane  and 
parallel  sailing,  and  is  used  for  reducing  the  departure  to 
miles  of  longitude.  Now,  when  two  places  lie  not  on  the 
same  parallel,  their  difference  of  longitude,  reduced  to 
miles  of  easting,  or  westing,  if  reckoned  on  the  higher 
parallel,  would  be  too  small,  and  if  on  the  lower  parallel, 
too  great.  The  common  way  of  reducing  it  is,  by  taking 
it,  as  the  name  implies,  on  a  parallel  midway  between 
the  two;  which,  though  not  strictly  correct,  is  sufficiently 
so  for  most  nautical  purposes.  For  the  solution  of  ques- 
tions of  this  kind,  we  have  only  to  place  together  the  two 
triangles  treated  of  under  Plane  and  Parallel  Sailing,  and 
resolve  them  separately,  observing  to  begin  with  that  in 
which  two  parts  are  given,  and  then  the  unknown  parts 
of  the  other  triangle  will  be  easily  obtained.  See  triangle 
ACK,  (diagram,  page  48.)  By  plane  sailing,  the  angle 
at  K  is  the  course ;  KD,  the  difference  of  latitude ;  DA, 
the  departure ;  and  KA,  the  distance  sailed.  By  parallel 
sailing  AD  is  still  the  departure,  or  meridian  distance,  on 
the  parallel  midway  between  the  latitude  left  and  latitude 
reached ;  CAD,  the  angle  of  the  middle  latitude ;  and 
AC,  the  difference  of  longitude.  The  following  examples 
will  illustrate  the  modes  of  solution. 


EXAMPLES. 

21.  Required  the  course  and  distance  from  the  east 
point  of  St.  Michael's,  in  lat.  37°  49'  N.,  long.  25°  11' 
W.,  to  Start  Point,  in  lat.  50°  13'  N.,  long.  3°  38'  W. 


192  A  TREATISE   ON   A  BOX   OP 

50°  13'  N.  25°  11'  W. 

37   49  N.  3    38  W 


J)12   24  =  744,  diff.  lat.      21    33  =  1293,  diff.  long. 

6    12 

37   49 


44      1  mid.  lat.,  complement  of  which  =  45°  59'. 


Then  sin.  90°  :  1293  :  :  sin.  45°  59' :  930,  the  departure. 

744  :  rad. : :  930 :  tan.  51°  20',  the  course  =  N.  51°  20'  E. 

sin.  51°  20' :  930  :  :  sin.  90°  :  1191  miles,  the  distance. 

22.  A  ship  from  Brest,  in  lat.  48°  23'  N.,  long.  4°  30' 
W.,  sailed  S.W.  f  W.  238  miles :  required  the  lat.  and 
long,  come  to. 

S.W.  i  W.  ==  53°  26',  comp.  of  which  =  36°  34'  j 
Then 
sin.  90°  :  238  :  :  sin.  36°  34'  :  141.8  diff.  lat.  ==  2°  22'. 

48°  23'  N. 
2   22 

46      1  lat.  come  to. 
1    11  =  *  diff.  lat. 


47    12  mid.  lat.,  comp.  of  which  —  42°  48'. 

Then  sin.  42°  48'  :  238  :  :  sin.  53°  26'  :  282  diff,  long. 

=  4°  42'  W. 
4   30  W. 


9    12  long,  come  to. 


INSTRUMENTS   AND   THE   SLIDE-RULE.  193 

23.  A  ship  from  lat.  17°  N.,  long.  24°  25'  W.,  sailed 
N.W.  |  N.  till,  by  observation,  her  lat.  is  found  to  be 
28°  34'  N.  :  required  the  distance  sailed  and  long, 
come  to. 

N.W.  *  N.  =  36°  34',  comp.  of  which  =  53°  26'. 

28°  34'  N. 
17     0  N. 


34  =  694  diff.  lat. 


5   47 
17     0 

22    47  mid.  lat.,  comp.  of  which  =  67°  13'. 

Then 
sin.  53°  26' :  694  :  :  sin.  90°  :  864  miles,  the  distance; 

and  sin.  67°  13'  :  864  :  :  sin.  36°  34'  :  558,  diff.  long. 
=  9°  18'  W. 
24   25  W. 


33   43  long,  come  to. 


EXAMPLES   FOR   PRACTICR 

24.  A  ship  from  lat.  26°  30'  N.,  long.  45°  30'  W., 
sailed  N.E.  £  N.  till  her  departure  was  216  miles:  re- 
quired the  distance  run,  and  lat.  and  long,  come  to. — 
Ans.  Dist.  341  miles;  lat.  come  to,  30°  53'  N.;  long. 
41°  24'  W. 

25.  From  lat.  43°  24'  N.,  long.  65°  39'  W.,  a  ship 
sailed  246  miles,  on  a  direct  course  between  S.  and  E., 
and  was  then,  by  observation,  in  lat.  40°  48'  N. ;  required 


194  A   TREATISE   ON   A   BOX   OF 

the  course,  and  long.  in. — Ans.    Course,  50°  40';  long 
come  to,  61°  23'  W. 

26.  A  ship  from  Cape  St.  Vincent,  lat.  37°  2'  N.,  long. 
9°  2'  W.,  sails  between  S.  and  W.;  the  lat.  come  to  ia 
18°  16r  N.,  and  departure  838  miles :  required  the  course, 
distance  run,  and  long,  come  to. — Ans.  Course,  36°  40' j 
dist.  1403  miles ;  long,  come  to,  24°  48'  W. 

27.  A  ship  from  Bordeaux,  in  lat.  44°  5(K  N.,  0°  35' 
W.,  sails  between  the  N.  and  W.  374  miles,  and  makes 
210  miles  of  easting :  required  the  course,  and  lat.  and 
long,  come  to. — Ans.  Course,  34°  10' ;  lat.  come  to,  49° 
59'  N.,  long.  5°  45'  W. 

28.  A  ship  from  lat.  54°  56'  N.,  long.  1°  10' W., 
sailed  between  N.  and  E.  till,  by  observation,  she  was 
found  to  be  in  long.  5°  26'  E.,  and  had  made  220  miles 
of  easting:    required  the  lat.  come  to,  and  course  and 
distance  run. — Ans.   Lat.  come  to,  57°  34'  N. ;  course, 
54°  20';  distance,  271  miles. 

29.  A  ship  from  a  port  in  N.  lat.  sailed  S.  E.  i  S.  438 
miles,  and  differed  her  long.  7°  28':    required  the  lat. 
sailed  from  and  come  to. — Ans.  Lat.  sailed  from,  51°  40' j 
come  to,  46°  16 . 


INS1RUMENTS   AND  THE   SLIDE-RULE.  195 


TO  DETERMINE  THE  DIFFERENCE  OF  LONGITUDE 
MADE  GOOD  UPON  COMPOUND  COURSES,  BY 
MIDDLE  LATITUDE  SAILING. 

WITH  the  several  courses  and  distances  find  the  latitude 
and  departure  made  good  and  the  ship's  present  latitude, 
as  in  Traverse  Sailing.  Take  the  middle  latitude  between 
the  latitude  left  and  latitude  arrived  at;  then  with  the 
departure  made  by  the  traverse  table,  and  the  middle  lati- 
tude, find  the  difference  of  longitude  by  Middle  Latitude 
Sailing.  In  high  latitudes  this  method  will  be  somewhat 
incorrect,  and  therefore  it  will  be  advisable  to  employ  the 
more  tedious  mode  of  computing  the  difference  of  longi- 
tude for  every  separate  course,  which  is  most  readily  done 
as  follows : — Complete  the  traverse  table,  as  before,  to 
which  annex  five  columns  :  in  the  first  put  the  several  lati- 
tudes the  ship  is  in  at  the  end  of  each  course  and  distance ; 
in  the  second,  the  sums  of  each  consecutive  pair  of  lati- 
tudes ;  and  in  the  third,  half  the  sums,  or  middle  latitude  ; 
then  find  the  difference  of  longitude  answering  to  each 
separate  middle  latitude,  and  its  corresponding  departure, 
and  place  it  in  the  fourth  or  fifth  (namely  the  east  or 
west)  difference  of  longitude  columns,  according  as  the  de- 
parture is  east  or  west :  then  the  difference  of  the  sums 
of  the  east  and  west  columns  will  be  the  difference  of  longi- 
tude made  good,  of  the  same  name  as  the  greater. 


196 


A   TREATISE   ON   A  BOX   OF 


EXAMPLE. 


30.  A  ship  from  lat.  66°  14'  N.,  long.  3°  12'  E.,  sails 
NN.  E.  *  E.  46,  N.  E.  i  E.  28,  N.  I W.  52,  N.  E.  b.  E.  i  E. 
57,  and  E.  S.  E.  24  miles  :  required  her  course,  and  longi- 
tude in. 


to 

1 

* 

MSN 

8      j                     | 

= 

< 

<O  •*>         COiO 
3'  »O     t   <N  O 
>o   '  co  3 

S--  -                 I 

§*    ^             ^a    3 

8[pp 

P«T 

!I« 

CO         COrH  <T» 

S3    g 

i 

Iml 

J           1 

5» 

rjBT 

9  IB  &  3  3  3 

11 

i 

11511 

5 

0 

H 

s^isa 

„»  s                -•«•§§ 
T**-1    »'                   3    s  s  r- 
S       S                      g,    s  -2  S 

3 

< 

1  I  I  12 

fei                       S    ^a  or  te            x 

«     o            i  8  *  s  „         -S 

&         3  i  |  ^  u       1 

ti 

« 

5rH  SiM    1 

I3    1    riri-lfl  |J4  rfri   f 

cf  GC  C^l  t~  H* 

•*  cS  o  S  cS 

|«  «-   B  IISVI- 

Course. 

H"      ^- 

*5  -g                       g  gj  8,     M  J 

"Sis               g  .c  _c  3S 

INSTRUMENTS   AND   THE    SLIDE-RULE.  197 


EXAMPLES   FOR   PRACTICE. 

31.  If  a  ship  sail  from  the  Naze,  in  lat.  57°  58'  N., 
long.  7°  3'  E.,  W.  X.  W.  24,  X.  W.  *  W.  16,  SS.  W.  31, 
S.  J  E.  12,  and  S.  W.  f  S.  20  miles :  required  her  lat.  and 
long.— Ans.  Course  S.  56°  24'  W.;  lat.  57°  20' X.;  loug. 
5°  15'  E. 

32.  If  a  ship  sail  from  the  Cape  of  Good  Hope,  lat. 
34°  29'  S.,  18°  23'  E.,  N.  W.  25,  N.  J  W.  21,  NX.  E. 
i  E.  35,  N.  W.  |  W.  40,  and  N.  b.  E.  18  miles  :  required 
her  lat.  and  long.— Ans.  Lat.  32°  37'  S.,  and  long.  17° 
43' E. 


MERCATOR'S  SAILING. 

IN  Mercator's  Sailing,  so  called  from  the  name  of  its 
inventor,  Gerard  Mercator,  the  earth  is  conceived  to  be 
projected  on  a  plane.  In  this  projection,  the  meridians 
are  parallel  to  each  other,  and,  consequently,  all  places 
upon  it  are  distorted,  and  the  more  so  as  they  approach 
the  poles;  but,  to  compensate  for  this  distortion,  the  de- 
grees of  latitude  are  everywhere  increased  in  the  same 
proportion  as  those  of  longitude ;  and,  consequently,  the 
bearings  between  places,  and  the  proportions  between  the 
latitude,  longitude,  and  nautical  distance,  will  be  the  same 
as  those  on  the  globe.  To  examine  into  this  proportion, 
Jet  us  refer  again  to  diagram,  page  48.  It  was  shown,  in 
parallel  sailing,  that  any  arc  described  by  C  is  to  a  similar 
arc  described  by  B  as  AD  to  AC.  But  AD  :  AC  :  : 
AB  :  AG;  consequently,  any  arc  described  by  C  is  to 
any  similar  arc  described  by  B  as  AB  is  to  AG  ;  that  is, 

17* 


198 


A   TREATISE   ON   A   BOX   OF 


as  radius  is  to  the  secant  of  the  latitude.  If,  therefore,  as 
in  Mercator's  Projection,  the  meridians  are  everywhere 
equidistant,  and,  consequently,  each  parallel  of  latitude 
equal  to  the  equator;  then  the  length  of  any  arc,  as  of  a 
minute,  or  a  degree,  on  any  parallel,  is  elongated  beyond 
its  just  proportion,  in  the  same  ratio  as  the  secant  of  the 
latitude  of  that  parallel  exceeds  radius.  Therefore,  to 
keep  up  the  proportion  of  northing  and  southing  with  that 
of  easting  and  westing,  the  length  of  a  minute  upon  the 
meridian  at  any  parallel  must  be  increased  beyond  its  just 
proportion  in  the  ratio  of  the  secant  to  radius.  Conse- 
quently, the  meridional  parts  of  any  given  latitude  are 
found  by  adding  together  the  natural  secants  of  successive 
minute  portions  of  that  latitude ;  and  the  smaller  these 
are  taken,  the  more  correct  will  be  the  table  so  formed. 
One  sufficient  for  the  purposes  of  the  Slide -Rule  is  here 
given. 

TABLE  OF   MERIDIONAL  PARTS   TO   EVERY   DEGREE   OF 
THE   QUADRANT. 


0  MP. 

0 

MP. 

o 

MP. 

o 

MP. 

0 

MP. 

0 

MP. 

0 

MP. 

0 

PM. 

0 

MP. 

0   0 

10 

603 

20 

1225 

30 

1888 

40 

2623 

60 

3474 

60 

4527 

70 

5966 

80 

8375 

1   60 

11 

664 

21 

1289 

31 

1958 

41 

270'2 

ol 

3o«;;i 

61  4649 

71 

6146 

SI 

8739 

2  120 

12 

725 

22 

1354 

32 

2028 

42 

2782 

.">2 

36C5 

62  4770 

72 

6335 

82 

9145 

3  180 

13 

787 

23 

1419 

33 

2100 

43 

2863 

58 

3764 

•334905 

73 

6534 

S3 

9606 

4  240 

14 

848 

24 

1484 

34 

2171  44 

2946 

54 

3865 

645039 

74 

6746 

84 

10137 

5  300 

In 

910 

26 

155035 

2244  45 

3030 

.1.1 

396S 

65  '5179 

75 

6970 

80 

10765 

6  361 

If, 

973 

26 

161636 

2318i  46 

31  1C, 

.ifi 

4074 

66 

5324 

76 

721^ 

80 

11532 

7  421 

17 

10*5 

27 

168437 

2393;  47 

3203 

n 

4183 

07 

5474 

1  1 

7467 

87 

12522 

8  482 

18 

1098 

28 

1751  :38 

2468  48 

3292 

58 

4294 

68 

5631 

78 

7745 

88 

13916 

9  542 

I'l 

llfil 

29 

1819  39 

2545  49 

33S2 

.V.I 

4409 

Ii9 

5795 

79 

8046 

88 

16300 

To  return  to  the  diagram.  Let  the  angle  DAC  be  the 
course;  AD  the  difference  of  latitude;  AC  the  distance; 
and  DC  the  departure;  then  AB  being  the  elongated,  or 
meridional,  difference  of  latitude,  AG  will  be  the  elon- 


INSTRUMENTS   AND   THE   SLIDE-RULE.  199 

gated  distance,  and  BG  the  elongated  departure,  that  is 
the  real  difference  of  longitude. 

Now,  AD  :  DC  :  :  AB  :  BG;  that  is,  Diff.  Lat. 
:  Dep.  :  :  Merid.  Diff.  Lat.  :  Diff  Long. 

And  AB  :  BG  :  :  Had.  :  tan.  CAD ;  that  is,  Merid. 
Diff.  Lat.  :  Diff.  Long.  :  :  Rad.  :  tan.  course. 

To  find  the  meridional  parts  answering  to  any  number 
of  degrees  and  minutes,  take  proportional  parts  of  the  dif- 
ferences found  by  subtraction. 

EXAMPLES. 

33.  Required  the  meridional  parts  answering  to  37°  43', 
viz.  37g. 

38°  =  2468 
37°  =  2393 


75  Then  2393 

43  +      54 


225                =  2447  merid.  parts  for  37°  43'. 
300  


6,0  )  322,5 


54    nearly. 


34.  Required  the  meridional  parts  for  27°  58'. 
28°  =  1751 

27°  =  1684            Then  1751 
2 


67 


=  1749  merid.  parts  for  27°  58'. 
6,0  )  13,4 


200  A   TREATISE   ON   A  BOX   OF 

35.  Required  the  number  of  degrees  answering  to  3625 
meridional  parts. 

3665  =  52°  3625 

3569  =  51°  3569 


96=    I°or60'  56=?' 


96   :   60'  :  :  56  :  35'. 

.-.  3625  =  51°  35'. 

EXAMPLES  IN   MERCATOB/S    SAILING. 

36.  Required  the  course  and  distance  from  the  east 
point  of  the  Azores,  lat.  37°  49'  N.,  long.  25'  11'  W.,  to 
Start  Point  in  lat.  50°  13'  N.  long.  3°  38'  W. 

25°  11'  W,  60°  13'  N.  Meridional  parts  =  3495 

3    38  W.  37  49  N.  "  =  2454 


21  33=1293  miles  12  24  =  744  miles  diff.  lat.     1041  merid. 
diff.  long diff.  lat. 

Then  1041  :  rad.  :  :  1293  :  tan.  51°  10'  course,  whose 
comp.  =38°  50'; 

and  sin.  38°  507  :  744  :  :  sin.  90°  :  1186,  the  distance. 
Compare  this  with  Example  21. 

37.  A  ship  sails  from  lat.  38°  47'  N.,  long.  75°  4^  W.,  267 
miles  N.  E.  b.  N. :  required  the  ship's  present  place. 

N.  E.  b.  N.  =  33°  45'  course  :  comp.  of  which=56°15' 
Bin.  91°:  267  : :  sin.  56°  15'  :  222  diff.  lat.=  3°42'N. 

38  47  N. 


42   29      laL 
come  to 

42°  29'  merid.  parts  =  2821 

38°  47'          "          —  2528 


293  merid.  diff.  lat 


INSTRUMENTS  AND   THE   SLIDE-RULE.  201 

Rad.  :  293  : :  tan.  33°  45'  :  196  diff.  long.  =3°  16'  E. 

75°    4'W. 
3    16  E. 


71    48  long.  in. 

38.  A  ship  from  Nova  Scotia,  in  lat.  45°  20'  N.,  long. 
60°  55'  W.,  sailed  S.  E.  |  S.,  and,  by  observation,  was 
found  to  be  in  lat.  41°  14'  N. :  required  the  distance 
sailed,  and  long,  come  to. 

S.  E.  i  S.  =  42°  11'  course,  whose  comp.  =47°  49'. 
45°  20'  merid.  parts  =  3058 
41   14        do.  =2720 


4     6  =  diff.  lat.  338  merid.  diff.  lat. 

liad.  :  338  :  :  tan.  42°  11'  :  306  diff.  long.  =  5°  6' 
60°  55'  W. 
5     6  E. 


55   49  W.  long.  in. 
Sin.  47°  49'  :  246  :  :  sin.  90°  :  332,  distance. 

EXAMPLES   FOR   PRACTICE. 

39.  Required  the  direct  course  and  distance  between 
the  Lizard  in  lat.  50°  0'  N.,  and  Port  Royal,  in  Jamaica, 
in  lat.  17°  40'  N.,  differing  in  long.  70°  46',  Port  Royal 
lying  so  far  to  the  W.  of  the  Lizard. — Ans.  Course,  60° 
33';  distance,  3645  miles. 

40.  Suppose  a  ship  from  the  Lizard,  in  lat.  50°  N.,  sails 
S.  35°  40'  W.  156  miles :  required  lat.  come  to,  and  how 
much  she  has  altered  her  longitude. — Ans.  Lat.  come  to, 
47°53'N.;  diff.  long.  2°  19'. 


202  A   TREATISE   ON  A  BOX  OP 

41.  A  ship  in  lat.  54°  20'  N.  sails  S.  33°  45'  E.,  until, 
by  observation,   she  is  found  to  be  in  lat.  51°  45'  N. : 
required  the  distance  sailed,  and  the  diff.  long. — Ans. 
Distance,  186.4  miles  :  diff.  long.  2°  52'  E. 

42.  A  ship  from  lat.  45°  26'  N.  sails  between  N.  and  E. 
195  miles,  and  then,  by  observation,  is  found  to  be  in  lat. 
48°  6'  N. :  required  the  direct  course,  and  diff.  long. — 
Ans.  Course,  N.  34°  52'  E.,  or  N.  E.  b.  N.  1°  7'  E. ;  diff. 
long.  2°  43'  E. 

43.  A  ship  from  lat.  48°  50'  N.  sails  S.  34°  40'  E.,  till 
her  diff.  long,  is  2°  44' :  required  lat.  come  to,  and  dis- 
tance sailed. — Ans.  Diff.  lat.  2°  41' ;  distance  196  miles. 

44.  A  ship  from  54°  36'  N.  sails  S.  42°  33'  W.,  until 
she  has  made  116  miles  of  departure :  required  the  lat. 
she  is  in,  her  direct  distance  sailed,  and  how  much  she 
has  altered  her  longitude. — Ans.  Lat.  come  to  52°  30' ; 
distance,  171.5  miles;  diff.  long.  3°  15'. 


INSTRUMENTS   AND   THE   SLIDE-RULE.  203 


TO  DETERMINE  THE  DIFFERENCE  OF  LONGITUDE 
MADE  GOOD  UPON  COMPOUND  COURSES,  BY 
MERCATOR'S  SAILING. 

With  the  several  courses  and  distances,  find  the  latitude 
and  departure  made  good,  and  the  ship's  present  latitude, 
as  in  traverse  sailing.  Take  the  meridional  difference  of 
latitude  between  the  latitude  left  and  latitude  arrived  at. 
Then,  with  the  course  made  good  by  the  traverse  table, 
and  the  meridional  difference  of  latitude,  find  the  difference 
of  longitude  by  Mercator's  Sailing.  In  high  latitudes, 
this  method  will  be  somewhat  incorrect;  and,  therefore, 
it  will  be  advisable  to  employ  the  more  tedious  mode  of 
computing  the  difference  of  longitude  for  every  separate 
course,  which  is  most  readily  done  as  follows  : — Complete 
the  traverse  table  as  before,  to  which  annex  five  columns. 
In  the  first,  put  the  several  latitudes  the  ship  is  in  at  the 
end  of  each  course ;  in  the  second,  the  meridional  parts 
corresponding  to  each  latitude;  and  in  the  third,  the  dif- 
ference of  each  consecutive  pair  of  meridional  parts. 
Then  find  the  difference  of  longitude  answering  to  each 
separate  course,  and  its  corresponding  meridional  differ- 
ence of  latitude,  and  place  it  in  the  fourth  or  fifth  (viz. 
the  east  or  west)  difference  of  longitude  columns,  according 
as  the  course  is  east  or  west;  then  the  difference  of  the 
sums  of  the  east  and  west  columns  will  be  the  difference 
of  longitude  made  good,  of  the  same  name  as  the  greater. 

EXAMPLE. 

45.  A  ship  from  lat.  66°  14' N.,  long.  3°  12'  E.,  sails 
NN.  E.  i  E.  46,  N.  E.  }  E.  28,  N.  }  W.  52,  N.  E.  b.  E.  t 


204 


A   TREATISE   ON   A  BOX   OP 


E.  57,  and  E.  S.  E.  24  miles :   required  her  course  and 
longitude  in. 


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EXAMPLES   FOR  PRACTICE. 


46.  A  ship  from  lat.  57°30'N.,  long.  1°47'W.,  sailed 
SS.  E.  48,  S.  W.b.  S.  54,  E.  b.  S.  71,  N.  E.  63,  and  W.  N. W 
50  miles  :  required  the  lat.  and  long,  of  the  place  come 
to. — Ans.  By  1st  rule,  lat.  come  to,  56°  50'  N.,  long 
2' W. 


INSTRUMENTS   AND   THE   SLIDE-RULE.  205 

47.  Four  days  ago,  we  took  our  departure  from  Faro- 
head,  in  lat.  58°  4(X  N.,  and  long.  4°  50'  W.,  and  since 
have  sailed  as  follows :— N.W.  32,  W.  69,  W.  N.  W.  93, 
W.  b.  S.  77,  S.W.  58,  and  W.  f  S.  49  miles :  required 
our  present  lat.  and  long. — Ans.  By  Rule  2,  lat.  come  to, 
58°  35';  long.  15°  54' W. 


OBLIQUE  SAILING. 

Oblique  Sailing  is  the  application  of  oblique  angled 
plane  triangles  to  the  solution  of  problems  at  sea ;  and  is 
particularly  useful  in  going  along  shore,  and  surveying 
coasts  and  harbours. 

EXAMPLES. 

48.  Coasting  along  the  shore,  I  saw  a  cape  bear  from 
me  NX.  E. ;  then  I  stood  away  X.W.  b.  W.  20  miles,  and 
observed  the  same  cape  to  bear  from  me  N.  E.  b.  E. : 
required  the  distance  of  the  ship  from  the  cape  at  her  last 
station.     See  figure,  page  72. 

Sin.  33°  45'  :  20  ::  sin.  78°  45'  :  35.3  miles. 

49.  A  point  of  land  was  observed,  by  a  ship  at  sea,  to 
bear  E.  b.  S. ;  and  after  sailing  N.  E.  12  miles,  it  was 
found  to  bear  S.  E.  b.  E.     It  is  required  to  determine  the 
place  of  that  headland,  and  the  ship's  distance  from  it  at 
the  last  observation.     See  figure,  page  78. 

Sin.  22°  307  :  12  : :  sin.  56°  15'  :  26.1. 

50.  At  noon,  Dungeness  bore  N.  b.  W.,  distance  5 
leagues ;  and  having  run  N.  W.  b.  W.  7  knots  an  hour,  at 
5  P.  M.  we  were  up  with  Beachy  Head  :  required  the  dis- 
tance of  Beachy  Head  from  Dungeness. — Ans.  26.6  mile* 

u 


206  A   TREATISE   ON   A   BOX   OF 

WINDWARD  SAILING. 

Windward  Sailing  is  the  method  of  gaining  an  intended 
port  by  the  shortest  and  most  direct  method  possible,  when 
the  wind  is  in  a  direction  unfavourable  to  the  course  the 
ship  ought  to  steer  for  that  port.  In  order  to  attain  this 
point,  it  is  evident  that  the  ship  must  sail  on  different 
tacks;  and,  therefore,  the  object  of  this  sailing  is,  to  fiud 
the  proper  courses  to  be  steered  on  each  board,  that  the 
vessel  may  arrive  at  the  intended  port  with  the  least  delay 
possible.  By  the  term  board  is  to  be  understood  the 
shifting  of  the  direction,  or  alteration  of  the  course.  Thus, 
if  a  vessel  sails  on  two  boards,  she  shapes  out  the  letter 
V ;  if  on  three  boards,  the  letter  N ;  and  so  on. 

EXAMPLES. 

51.  A  ship  is  bound  to  a  port  48  miles  directly  to  the 
windward,  the  wind  being  SS.W.,  which  it  is  intended  to 
reach  on  two  boards ;  and  the  ship  can  lie  within  6  points 
of  the  wind ;  required  the  course  and  distance  on  each 
tack. 

Describe  a  circle,  and  from  the  centre,  which  call  A, 
draw  a  line  in  a  SS.W.  direction,  to  represent  the  direction 
of  the  wind,  and  call  the  lower  extremity  of  this  line  B,  and 
let  it  represent  the  port  intended  to  be  reached.  Then 
the  wind  blowing  from  B  to  A,  and  A  being  the  position 
of  the  ship ;  from  A,  to  the  left  of  the  line  BA,  draw  a 
line,  making  with  it  an  angle  of  6  points,  or  67°  30' ; 
this  will,  of  course,  be  due  W.  From  the  centre  of  the 
circle  A,  to  the  right  of  the  line  AB,  draw  another  line, 
making  with  AB  an  angle,  like  the  other,  of  67°  30'. 
This  line  will  be  south-east.  From  the  point  B,  paralle 


INSTRUMENTS   AND   THE   SLIDE-RULE.  207 

to  this  last  line,  draw  a  line,  cutting  the  one  running 
•west,  in  a  point,  which  call  C.  Then  AC  will  be  the 
course  of  the  ship  on  the  first  board,  and  CB  that  on  the 
second.  Now,  the  angles  at  A  and  B  will  be  each  67  J°, 
and  at  C  45°,  opposite  which  is  the  line  AB,  48  miles. 

Then,  sin.  45°  :  48  miles  : :  sin.  67  J°  :  62.7  miles,  the 
distance  to  be  sailed  on  each  board ;  so  that  she  will  have 
to  sail  125.4  miles  to  make  48. 

52.  The  wind  at  N.  W.,  a  ship  bound  to  a  port  64  miles 
to  the  windward,  proposes  to  reach  it  on  three  boards,  two 
on  the  starboard,  and  one  on  the  larboard  tack,  and  each 
within  5  points  of  the  wind  :  required  the  course  and  dis- 
tance on  each  tack. 

Describe  a  circle,  and  from  its  centre,  which  call  A, 
draw  a  line  in  a  N.  W.  direction,  to  represent  the  direc- 
tion of  the  wind,  and  let  its  upper  extremity  denote  the 
port  intended  to  be  reached,  which  call  B.  From  A  draw 
two  lines,  one  to  the  left  and  the  other  to  the  right  of  the 
line  B  A,  each  making  with  it  an  angle  of  5  points ;  con- 
sequently, the  first  will  pass  through  the  W.  b.  S.  rhumb, 
and  the  second  through  the  N.  b.  E.  Call  the  lower  ex- 
tremity of  the  line  passing  through  the  S.  b.  W.  rhumb, 
C ;  the  upper  extremity  of  the  other,  D.  From  B  draw 
a  line  to  the  right  of  the  line  BA,  parallel  with  CA.  Bi- 
sect BA,  in  a  point,  which  call  E.  Draw  a  line  from  E 
to  C,  parallel  with  the  line  DA,  and  prolong  it  upward 
till  it  cuts  the  line  running  right  of  B,  in  a  point,  which 
call  F.  Then,  in  the  triangle  EAC,  the  angles  at  A  and 
C  are  each  56°  15',  and  the  angle  at  E  67°  30',  and  the 
line  EA  is  32  miles.  Therefore,  sin.  56°  15'  :  32  : : 
sin.  67°  30'  :  36.25  miles  =  AC,  BF,  CK,  or  EF,  and 


208  A   TREATISE   ON   A   BOX   OP 

twice  36.25  =  72.5.  Hence,  she  must  first  sail  W  b.  S. 
36J  miles,  then  N.  b.  E.  72£  miles,  then  W.  b.  S.  36* 
miles. 

It  may  be  here  observed,  that  whatever  number  of 
boards  it  may  be  found  expedient  a  ship  should  make,  the 
sum  of  the  distances  on  each  tack  will  be  the  same  as  if 
the  place  had  been  reached  on  two  boards  only. 

53.  A  ship  is  bound  to  a  port  26  miles  directly  to  wind- 
ward (the  wind  being  N.  E.,)  which  it  is  intended  to  reach 
on  two  boards,  the  first  being  on  the  larboard  tack,  and 
the  ship  can  lie  within  6  points  of  the  wind  :  required  the 
course  and  distance  on  each  tack. — Ans.  Course  on  the 
larboard  tack,  E.  S.  E. ;  on  the  starboard,  NN.  W. ;  dis- 
tance on  each  board,  34  miles,  nearly. 

54.  The  wind  at  N.  £  E.,  a  ship  is  bound  to  a  port 
bearing  NN.  E.,  distance  68  miles,  which  it  is  proposed 
to  make  at  four  boards ;  the  coast,  which  is  to  westward, 
trends  NN.  E.  also ;  so  that  the  ship  must  go  about  as 
soon  as  she  reaches  the  straight  line  joining  the  ports : 
required  the  course  and  distance  on  each  board,  the  ship 
making  her  way  good  within  6  points  of  the  wind. — Ans. 
Course  on  the  larboard  tack,  E.  N.  E.  £  E. ;  on  the  star- 
board, N.  W.  b.  W.  J  W. ;  first  and  third  distances,  47.8 
miles;  second  and  fourth  distances,  37.2  miles. 

55.  A  ship  close  hauled  within  5  points  of  the  wind, 
and  making  1  point  of  leeway,  is  bound  to  a  port  bearing 
SS.  W.,  distant  54  miles,  the  wind  being  S.  b.  E. ;  it  is 
intended  to  make  the  port  at  three  boards,  the  first  of 
which  must  be  on  the  larboard  tack,  in  order  to  avoid  a 
reef  of  rocks :  required  the  course  and  distance  on  each 


INSTRUMENTS  AND  THE   SLIDE-RULE.  209 

tack. — Ans.  Course  on  the  larboard  tack,  S.  W.  b.  TV. ; 
on  the  starboard,  E.  b.  S. ;  distances  on  the  larboard 
tack,  each  37.45  miles;  distance  on  the  starboard  tack, 
42.4  miles. 


CURRENT   SAILING. 

When  a  ship  sails  exactly  with  the  current,  her  velocity 
will,  of  course,  be  accelerated;  and,  when  in  due  opposi- 
tion to  the  current,  it  will  be  retarded  by  the  difference 
of  the  velocities  of  the  wind  and  stream.  When  she  is 
urged  by  the  wind  in  one  direction,  aud  by  the  current  in 
another,  her  course,  agreeably  to  the  law  influencing  all 
bodies  acted  upon  simultaneously  by  two  forces,  will  lie 
in  the  diagonal  of  the  parallelogram  formed  by  those  forces; 
that  is,  will  be  the  third  side  of  a  triangle  of  which  the 
drift  of  the  current  and  the  action  of  the  wind  form  the 
other  two,  the  angle  between  them  being  known. 

N.B.  That  point  of  the  compass  to  which  a  current  runs 
is  called  its  setting,  and  its  rate  per  hour  is  called  its  drift. 

EXAMPLES. 

56.  A  ship  sails  by  the  compass  directly  S.  96  miles, 
in  a  current  that  sets  E.  45  miles  in  the  same  time  :  re- 
quired the  ship's  true  course  and  distance. 

Describe  a  circle,  and  from  its  centre,  which  call  A, 
draw  a  line  in  a  south  direction,  and  make  it  equal  to  96 
from  a  scale  of  equal  parts,  and  call  the  lower  extremity  B. 
From  the  point  B,  in  an  easterly  direction,  draw  a  line 
equal  to  45,  from  the  same  scale,  and  call  its  extremity  C. 
Join  AC.  The  angle  BAG  will  be  the  course,  and  C  the 
point  at  which  the  ship  will  have  arrived.  Then, 

18* 


210  A   TREATISE  ON  A  BOX   OF 

96  :  rad.  : :  45  :  tan.  25°  7',  the  ship's  course  = 
SS.  E.  2°  6',  easterly. 

And,  sin.  25°  T  :  45  : :  sin.  90°  :  105.9  miles,  distance 
sailed. 

57.  A  ship  has  made  by  the  reckoning  N.  J  W.  20 
miles,  but,  by  observation,  it  is  found  that,  owing  to  a 
current,  she  has  actually  gone  NN.  E.  28  miles  :  required 
the  setting  and  drift  of  the  current  in  the  time  which  the 
ship  had  been  running. — Ans.  Setting,  N.  64°  48'  E., 
drift,  14.1  miles. 

58.  A  ship  from  a  port  in  lat.  42°  52'  N.,  sailed  S.  b. 
W.  £  W.  17  miles  in  7  hours,  in  a  current  setting  be- 
tween the  N.  and  W. ;  and  then  the  same  port  bore  E. 
N.  E.,  and  the  ship's  latitude,  by  observation,  was  42°  42' 
N. :  required  the  setting  and  drift  of  the  current. — Ans. 
Setting,  71°  55',  drift,  2.9  knots  an  hour. 

59.  A  ship,  bound  from  Dover  to  Calais,  lying  21  miles 
to  the  S.  E.  b.  E.  £  E.,  and  the  flood-tide  setting  N.  E.  £ 
E.  2 1  miles  an  hour :  required  the  course  she  must  steer, 
and  the  distance  run  by  the  log,  at  6  knots  an  hour,  to 
reach  her  port. — Ans.  Course,  39°  14'.     Distance  to  be 
run  19.4  miles. 

60.  From  a  ship,  in  a  current,  steering  W.  S.  W.  6 
miles  an  hour  by  the  log,  a  rock  was  seen  at  6  in  the 
evening,  bearing  S.  W.  i  S.  20  miles.    The  ship  was  lost 
on  the  rock  at  11  p.  M.  :  required  the  setting  and  drift 
of  the  current.— Ans.  Setting,  S.  75°  10'  E.,  drift  3.11 
miles  per  hour. 


INSTRUMENTS  AND   THE   SLIDE-RULE.  211 


OF  A  SHIP'S  JOURNAL. 

A  journal  is  a  register  of  transactions  occurring  on 
board  a  ship,  and  should  contain  a  particular  detail  of 
every  thing  relative  to  the  navigation  of  the  vessel — as 
the  courses,  winds,  currents,  &c. — that  her  situation  may 
be  known  at  any  instant  at  which  it  may  be  required. 
The  computations  made  to  determine  the  place  of  a  ship 
from  the  courses  and  distances  run  in  24  hours,  are  called 
a  day's  work ;  and  the  latitude  and  longitude  of  a  ship 
deduced  therefrom,  are  called  the  latitude  and  longitude 
in,  by  account,  or,  by  dead  reckoning,  in  contradistinction 
to  the  latitude  and  longitude  as  determined  by  observation. 
At  the  time  of  leaving  land,  the  bearing  of  some  known 
place  is  to  be  observed,  and  its  distance  found,  either  by 
observation,  or  by  taking  its  bearing  at  two  different 
times,  from  two  different  places,  and  determining  its  dis- 
tance accordingly.  The  log-book,  which  is  to  contain  a 
daily  transcript  from  the  log-board,  is  to  be  divided  into 
7  columns.  In  the  first,  put  the  hours;  in  the  second 
and  third,  the  knots  and  fathoms  sailed  per  hour ;  in  the 
fourth,  the  courses ;  in  the  fifth,  the  winds ;  in  the  sixth, 
the  leeway;  and  in  the  seventh,  any  remark  that  may  be 
thought  necessary.  It  is  better,  however,  to  omit  the 
leeway  column,  and,  on  transcribing  from  the  log-board, 
to  make  the  proper  allowance,  and  to  enter  the  amended 
courses  only,  in  the  log-book.  After  this,  allow  for  the 
variation,  and  bring  them  into  a  traverse  table.  Find  the 
ship's  distance,  difference  of  latitude,  and  departure,  by 
plane  sailing;  then,  by  Mercator's,  or  middle  latitude 


212 


A   TREATISE   ON   A   BOX   OF 


sailing,  find  the  difference  of  longitude,  and  enter  it  ac- 
cordingly. The  following  specimen  will  furnish  an  idea ; 
but  the  present  work  being  intended  principally  to  show 
the  instrumental  modes  of  computation,  the  student  is 
referred  to  works  exclusively  on  Navigation,  for  more  com- 
plete information  upon  the  subject. 


£  ™ 

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vation. 


(M  ^  CD  QO  O  C<l  ^1  Tt<  CD  CO     O     ?3 


The  departure  is  taken  from  the  Lizard,  at  10  A.  M. 
The  bearing  is.  N.  E.  J  E.,  distance  15  miles.     Now,  the 


INSTRUMENTS   AND   THE    SLIDE-RULE. 


213 


opposite  point  is  S.W.  J  W.,  and  the  variation,  2J  points, 
being  allowed  to  the  left  hand,  because  it  is  westerly, 
gives  SS.W.,  the  true  bearing  of  the  ship  from  the 
Lizard ;  so  it  will  be  SS.W.,  15  miles.  The  course  the 
ship  has  been  going,  is  W.  b.  N.,  which,  corrected  for 
variation,  is  W.  S.W.  £  W. ;  and  the  distance  run  from  10 
A.  M.  to  noon,  is  16  miles.  Now,  insert  these  in  a  tra- 
verse table,  as  under,  and  find  the  diff.  lat.  and  departure 
to  each  course  and  distance  by  plane  sailing.  Hence  the 
diff.  lat.  and  departure  made  good  will  be  obtained,  with 
•which  the  course  and  distance  from  the  Lizard  will  be 
determined.  Then,  with  the  departure  and  middle  lati- 
tude find  the  difference  of  longitude. 


TRAVERSE  TABLE. 


Diff.  Lat 

Departure. 

N. 

S. 

E. 

W. 

SS.W. 

15 

13.9 

5.7 

W.S.W.  JW. 

16 

4.6 

15.3 

S.  48°  4<y  W. 

28 

18.5 

21. 

Lat.  left  49°  57'  N. 
Diff.  lat.        18  S. 


49    39  lat.  in. 

9  =      diff.  lat 


49    48  mid.  lat.  =  comp.  40°  12', 


214 


A   TREATISE   ON  A   BOX  OF 


18.5  :  rad.  :  :  21  :  tan.  48°  40',  the  course. 
sin.  48°  40' :  21  :  :  sin.  90°  :  28,  the  distance. 
Bin.  40°  12' :  28  :  :  sin.  48°  40'  :  32.6  diff.  long.  =  33'  nearly. 

Long,  left  5°  15'  W. 
Diff.  long.       33  W. 

6    48  W.  long,  come  to. 


fe- 


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a,  — • 


II 


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INSTRUMENTS   AND    THE   SLIDE-RULE. 


215 


The  several  courses  being  corrected  for  variation,  the 
diff.  lat.  and  departure,  answering  to  each  course  and 
distance,  will  be  as  under. 


Diff.  Lat 

Departure. 

N. 

S. 

E. 

W. 

S.W.  J  W. 

50 

31.7 

38.6 

S.W.  J  S. 

108 

83.5 

68.5 

SS.W.  J  W. 

66 

49.4 

26.4 

S.  38°52'W. 

212 

164.6 

133.5 

Yesterday's  lat.  49?  39'  N. 
Diff.  lat 2    45  S. 


46    54  N.  lat.  in. 
1    22  =     diff.  lat. 


48    16  mid.  lat.  =  comp.  41°  44'. 

165  :  rad. :  :  133  :  tan.  38°  52',  the  course. 

Sin.  38°  52'  :  133  :  :  sin.  90°  :  212,  the  distance. 

Sin.  41°  44' :  212  :  :  sin.  38°  52' :  200  diff.  long.  =  3°  20*. 

Yesterday's  long.  5°  48'  W. 
Diff.  long 3    20  W. 

9     8  W.  long,  in  by  account. 


216  A  TREATISE   ON   A  BOX   OF 


RECAPITULATION. 


QUESTIONS  ON  TABLE  I. 

1.  THE  diameter  of  a  circle  is  9  inches :  what  is  the 
circumference  ? — Ans.  28.27. 

2.  What  is  the  side  of  an  equal  square  ? — Ans.  7.97. 

3.  The  circumference  of  a  circle  is  23  inches :  what  is 
the  diameter  ? — Ans.  7.32. 

4.  What  is  the  side  of  an  inscribed  square? — Ans.  5.17. 

5.  The  side  of  a  square  is  18  :  what  is  the  diameter  of 
an  equal  circle  ? — Ans.  20.3. 

6.  What  is  the  circumference  of  an  equal  circle? — 
Ans.  63.8. 

7.  The  area  of  a  circle  is  24 :  what  is  the  area  of  its 
inscribed  square  ? — Ans.  15.27. 

8.  The  area  of  a  square  is  24  :  what  is  the  area  of  its 
inscribed  circle? — Ans.  18.85. 

QUESTIONS  ON  TABLE  II. 

9.  The  diameter  of  a  circle  is  25  inches :  what  is  the 
side  of  an  inscribed  equilateral  triangle? — Ans.  21.65. 

10.  Of  an  inscribed  pentagon? — Ans.  14.69. 

11.  Of  a  circumscribed  decagon? — Ans.  8.12. 

12.  Of  an  inscribed  undecagon  ? — Ans.  7.04. 

13.  Of  a  circumscribed  dodecagon  ?— Ans.  6.69. 

14.  The  diameter  of  a  circle  is  12  inches :  what  is  the 
Bide  of  a  square  inscribed  in  it? — Ans.  8.48. 


INSTRUMENTS   AND    THE    SLIDE-RULE.  217 

15.  A  circle,  whose  diameter  is  9  J  inches,  has  a  regular 
hexagon  surrounding  it :  what  is  the  length  of  each  side  ? 
— Ans.  5.33. 

16.  An  octagonal  tower  measures  7  feet  along  each 
side  :  what  will  be  the  diameter  of  a  circle  surrounding 
it?— Ans.  18.3. 

17.  What  is  the  length  of  the  longest  line  that  can  he 
drawn  within  a  dodecagon,  each  of  whose  sides  is  7  feet  ? 
—Ans.  26.13. 


QUESTIONS  ON  TABLE  III. 

18.  The  side  of  an  equilateral  triangle  is  7  :  what  is 
the  area  ?— Ans.  21.2. 

19.  The  side  of  a  regular  pentagon  is  7  :  what  is  the 
area?— Ans.  84.3. 

20.  The  side  of  a  regular  heptagon  is  7 :  what  is  the 
area? — Ans.  178. 

21.  The  side  of  a  regular  nonagon  is  7 :  what  is  the 
area?— Ans.  302.9. 

22.  The  side  of  a  regular  dodecagon  is  6  :  what  is  the 
area? — Ans.  403. 

23.  The  side  of  a  regular  hexagon  is  47  inches :  how 
many  square  feet  does  it  contain  ? — Ans.  39.86. 

24.  What  is  the  area,  in  square  yards,  of  an  undecagon 
whose  side  measures  17.9  feet  ? — Ans.  334. 


QUESTIONS  ON  TABLE  IV. 

25.  A  bullet,  let  fall  from  a  balloon,  was  half  a  minute 
before  it  struck  the  earth  :  how  high  was  the  aeronaut  at 
the  moment  it  was  dropped  ? — Ans.  4825  yds.,  or  2f  miles. 

26.  When  the  same  balloon  had  attained  an  altitude  of 
4  miles,  or  7040  yards,  another  bullet  was  let  fall :  how 
many  seconds  was  it  in  descending? — Aus.  36}  seconds. 


218  A  TREATISE   ON   A   BOX   OF 

27.  What  is  the  height  of  a  precipice,  if  a  stone  la 
7  seconds  in  falling  from  the  top  to  the  bottom  ? — Aus. 
788  feet. 

28.  A  string,  with  a  bullet  at  the  end,  being  suspended 
from  a  hook  in  the  ceiling,  is  found  to  vibrate  64  times 
per  minute :  what  is  the  distance  from  the  hook  to  the 
centre  of  the  bullet  ? — Ans.  34.4  inches. 

29.  How  often  will  a  pendulum,   100   inches   long, 
vibrate  per  minute? — Ans.  37  j  times. 

30.  A  revolving  pendulum  shapes  out  52  cones  in  a 
minute  :  determine  its  length  ? — Ans.  13  inches. 

31.  The  diameter  of  a  circle  is  60  inches :  what  is  the 
area? — Ans.  2827  J  square  inches. 

32.  The  diameter  of  a  sphere  is  9  inches :  what  is  the 
convex  surface  ? — Ans.  25.45  square  feet. 

33.  The  circumference  of  a  sphere  is  12  inches :  what 
is  the  surface? — Ans.  45.8  square  inches. 

34.  What  is   the   diagonal  of  a   square   whose   side 
measures  15.3  inches  ? — Ans.  21.63. 

35.  A  cube  measures  9  inches  along  the  side :  what 
will  be  the  diagonal  of  the  face,  and  what  of  the  cube  ? — 
Ans.  12.72  diagonal  of  the  face;  15.58  diagonal  of  the 
cube. 

QUESTIONS  ON  TABLE  V. 

36.  The  diameter  of  a  circle  is  9  inches :  how  many 
square  inches  does  it  contain? — Ans.  63.6. 

37.  The  diameter  of  a  circle  is  44  inches :  how  many 
square  feet  does  it  contain? — Ans.  10.55. 

38.  The  side  of  a  square  is  17.5  inches:  required  the 
area  in  square  feet. — Ans.  2.126. 

39.  The  diameters  of  an  ellipse  are  12  and  10  feet :  re- 
quired the  area  in  square  yards. — Ans.  10.47. 


INSTRUMENTS   AND   THE    SLIDE-RULE.  219 

40.  What  is  the  area  in  square  yards  of  a  cycloid, 
whose   generating  circle   has   a  diameter  of  3  feet  ? — 
Ans.  2.356. 

41.  Required  the  surface  of  a  cylinder,  in  square  feet, 
the  circumference  of  which  is  29  inches,  and  height  42 
inches. — Ans.  8.46. 

42.  The  diameter  of  a  sphere  is  73  feet :  what  is  the 
surface  in  square  rods? — Ans.  61.5  nearly. 


QUESTIONS  ON  TABLE  VI. 

43.  A  vessel  in  the  shape  of  a  square  prism  is  40  inches 
deep,  and  12  inches  square  :  how  many  solid  feet  does  it 
contain  ?— Ans.  3.33,  or  3J  feet. 

44.  An  inverted  octagonal  pyramid  measures  5  inches 
along  each  side  at  the  top,  and  is  13  inches  deep :  how 
many  gallons  will  it  contain  ? — Ans.  1.88  gallons. 

45.  A  dodccagonal  pyramid  measures  6  inches  along 
each  side  at  the  bottom,  and  is  16.6  inches  high :  how 
many  solid  feet  does  it  contain  ? — Ans.  1.29  feet. 

46.  A  cone  of  ice  is  50  inches  in  perpendicular  height; 
the  diameter  of  its  base  is  10  inches  :  required  its  weight. 
—Ans.  43.9  Ibs. 

47.  A  hollow  sphere,  11  inches  in  diameter,  is  filled 
with  tallow  :  required  its  weight. — Ans.  23.1  Ibs. 

48.  How  much  gunpowder  would   fill   the  same? — 
Ans.  23.4  Ibs. 

*  49.  The  axes  of  an  oblate  spheroid  are  20  and  22  inches : 
how  many  gallons  will  it  hold  ? — Ans.  18.2  gallons. 

50  The  axes  of  a  prolate  spheroid  are  20  and  22  inches  : 
how  many  gallons  will  it  hold? — Ans.  16.6  gallons. 

51.  The  axes  of  an  elliptic  cylinder  of  solid  gold  arc  4 


220  A   TREATISE   ON   A   BOX   OP 

and  9  inches ;  its  depth  3  inches  :  what  is  its  weight  ? — 
Ans.  60  Ibs.  avoir. 

52.  The  axes  of  an  elliptic  cone  of  brass  are  4  and  9 
inches;  its  depth  8  inches:  required  its  weight. — Ans. 
22.85  Ibs. 

53.  A  paraboloid  of  zinc  is  14  inches  high;  the  radius 
of  its  base  5  inches :  required  its  weight. — Ans.  146  Ibs. 
nearly. 

54.  A  parabolic  spindle  of  silver  is  23  inches  long ;  its 
diameter  8  inches  :  what  is  its  weight  ? — Ans.  233.6  Ibs. 
avoir. 

55.  The  length  of  a  cask  is  45  inches,  the  bung  dia- 
meter 36,  and  the  head  diameter  30  inches :  required  the 
content  for  each  of  the  four  varieties. 

Ans.  1st  variety  148.37  gallons. 
2d       «       147.76      " 
3d       "       139.96      « 
4th      "       139.2        " 


QUESTIONS  ON  TABLE  VII. 

56.  A  sphere  of  platinum  weighs  32  Ibs. :  required  its 
diameter. — Ans.  4.28  inches. 

57.  A  solid  globe  of  gold  weighs  40  Ibs. :  what  is  its 
diameter? — Ans.  4.78  inches. 

58.  A  sphere  5  inches  diameter  is  filled  with  quick- 
silver :  required  its  weight. — Ans.  32.4  Ibs. 

59.  Required  the  diameter  of  a  pound  rocket. — Ans. 

1.07  in. 

t, 

60.  The  internal  diameter  of  rockets  is  usually  f  of 
their  external :  what  then  is  the  internal  diameter  of  a 
6  Ib.  rocket? — Ans.  2  inches. 

61.  A  globe  of  ice  weighs  10  Ibs.  :  required  its  dia- 
meter.— Ans.  8.3  inches. 


INSTRUMENTS  AND   THE   SLIDE-RULE.  221 


QUESTIONS  OX  TABLE  VIII. 

62.  A  sphere  contains  100  cubic  inches :  required  its 
diameter. — Ans.  5.75  inches. 

63.  The  solidity  of  a  sphere  is  180 :  what  is  its  diame- 
ter ?— Ans.  7. 

64.  What  is  its  circumference  ? — Ans.  22. 

65.  The  solidity  of  an  octahedron  is  9  :  what  is  the 
length  of  each  of  its  sides  ? — Ans.  2.68. 

66.  The  moon  is  distant  240  thousand  miles  from  the 
earth,  and  the  time  of  her  complete  revolution  is  about 
27$  days:  at  what  distance   would  she  go  round  in  a 
week  ? — Ans.  96  thousand  miles. 

67.  The  distance  of  Venus  is  68  millions  of  miles :  how 
many  weeks  does  she  consume  in  traversing  her  orbit  ? — 
Ans.  32  weeks. 

68.  Vesta  performs  her  revolution  in  about  47}  lunar 
months  :  required  her  distance. — Ans.  225  million  miles. 

69.  Juno  is  distant  253  millions  of  miles;  how  many 
days  are  consumed  in  her  revolution  ? — Ans.  1590. 

70.  At  what  distance  would  a  planet  require  to  be 
placed  to  revolve  round  the  sun  in  2  years  ? — Ans.  151 
millions  of  miles. 

71.  Ceres  is  distant  263,  Pallas  265  millions  of  miles 
from  the  sun :  how  many  years  is  each  employed  in  he; 
circuit? — Ans.  Ceres,  4.6  years  ;  Pallas,  4.67  years. 


72.  I  have  3  balls,  weighing  1  lb.,  2  Ibs.,  and  7£  Ibs. 
respectively ;  the  smallest  is  3  inches  diameter  :  required 
the  diameter  of  the  other  two. — Ans.  378,  and  5.87 
inches. 

7-1.   A  cone  weighing  74  Ibs.  is  24.6  inches  high,  and 


222  TREATISE   ON   A   BOX   OF    INSTRUMENTS. 

10  inches  diameter  at  the  base :  required  the  size  of  a 
similar  cone,  weighing  100  Ibs. — Ans.  27.19  inches  high; 
11.05  diam. 

74.  I  have  two  similarly  shaped  casks,  the  dimensions 
of  one  are,  length  54,  head  34.8,  bung  44.8,  and  middle 
diameter  83  inches  ;  the  other  holds  2f  times  as  much  : 
required  the  dimensions. — Ans.  L.  72.3; '-H.  46.6;  B. 
59.98;  and  M.  111.1. 

75.  Out  of  a  sheet  of  metal,  of  uniform  thickness,  a 
piece  is  cut  in  the  shape  of  a  regular  decagon,  each  of 
whose  sides  measures  7  inches ;  its  weight  is  found  to  be 
8i  Ibs. ;  a  similar  piece  is  cut  from  the  same  sheet,  and 
weighs  23f  Ibs. :  what  is  the  length  of  each  of  its  sides? 
— Ans.  11.83  inches. 

76.  Find,  by  the  rule,  the  cube  root  of  141. — Ans.  5.2. 

77.  The  frustum  of  a  nonagonal  pyramid  measures  3 
inches  along  each  side  at  top,  and  4  at  bottom,  and  the 
depth  is  10  inches;  into  this  I  put  a  sphere  of  brass 
weighing  f  of  what  the  water  required  to  fill  the  vessel 
would  weigh :    what  is  the  diameter  of  the  sphere  ? — 
Ans.  4.1  inches. 

78.  A  cast-iron  cannon-ball  weighs  38  Ibs., :  required 
its  circumference. — Ans.  20.4  inches. 

79.  A  tap  2  inches  in  diameter  will  empty  a  cask  in 
•)8  J  minutes  :  what  must  be  the  size  of  one  to  empty  it  in 
an  hour  and  53 J  minutes? — Ans.  1.373  inches. 

SO.  A  has  a  globe  of  lead  4  inches  diameter;  B,  a 
globe  of  copper  of  the  same  weight :  what  is  its  diameter  ? 
— Ans.  4.34  inches. 

81.  The  tinker,  mentioned  at  page  162,  succeeded  in 
making  a  similar  vessel  to  contain  20  gallons:  required  its 
dimensions. — Ans.  Depth,  13.51  inches;  bottom  diame- 
ter, 16.96;  top,  282/. 


APPENDIX. 

DURING  the  sale  of  the  first  few  hundred  copies  of  the 
present  impression,  it  has  been  found  that  the  omission 
of  the  Compass  has  proved  a  great  inconvenience ;  it  is, 
therefore,  now  supplied,  as  above. 

•  of  the  operations  of  the  Slide-Rule  have  been 
exhibited  at  pp.  91,  92,  &c.  The  principal  of  them  may 
be  more  concisely  shown  as  follows  : — Let  a  and  A  denote 
any  two  logarithmic  distances  taken  on  the  A  line;  b  and 
B  any  two  equal  distances  on  the  B  line ;  and  so  on 
Then  a  varies  as  b  as  c  as  d  *,  and  A  as  B  as  C  as  Z)* ; 
e  as  d',  and  E  as  IP ;  a*  as  e*  as  d6,  and  -4s  as  E*  &s  I*. 

From  these  an  immense  variety  of  combinations  may 
be  formed ;  some  of  them  more  curious  than  usefui.  The 
former  class  the  student  can  investigate  for  himself;  of 
the  latter  kind  the  following  are  of  constant  occurrence. 

(1.)         a  :  b  : :  A  :  B,  whence  B  = 


224  A   TREATISE  ON   A   BOX  OF 

Of  this  class  are  all  cases  of  simple  proportion,  in- 
cluding multiplication,  division,  and  many  formulae  for 
surfaces.  In  multiplication  a  being  unity,  in  division  A; 
and,  for  surfaces,  a  a  divisor,  b  length,  and  A  breadth. 

bd* 

(2.)  a  :  b  : :  d-  :  c.  whence  c  = 

a 

Of  this  class  are  the  formulae  for  surfaces  and  solids, 
when  divisors  are  used  instead  of  gauge  points.  For  sur- 
faces d  will  be  a  side,  or  diameter,  or  mean  proportional 
between  two  dimensions,  and  b  a  quantity  varying  with 
the  boundary  of  the  surface. 

Cft 

(3.)  a  :  b  : :  d"  :  c,  whence  b  =  -^ • 

Of  this  class  are  the  formulae  for  surfaces,  in  which  d 
is  a  gauge  point,  c  length,  and  a  breadth. 

cD~ 
(4.)      c  :<#»::  C :  D\  whence  C  =  -—• 

CL 

Of  this  class  are  the  formulae  for  accelerated  motion, 
and  for  exhibiting  the  relations  of  similar  plane  figures  to 
each  other ;  for  finding  the  areas  of  surfaces,  and  the  con- 
tent of  solids;  d  being  a  gauge  point:  c,  in  surfaces,  a 
variable  quantity — in  solids,  length,  height,  or  depth  ;  and 
D  a  diameter,  side,  or  mean  proportional  between  two 
dimensions. 

eD3 
(5.)        e:d*-.:E:D*,  whence  E  =  -^> 

Of  this  class  are  the  formulae  for  determining  the  di- 
mensions of  spheres  from  their  weight,  or  solidity;  and 
for  exhibiting  the  relations  of  similar  solids  to  each  other. 

A3 
(6.)       a3  :  e*  : :  A3  :  E2,  whence  E  =  ej/— 3- 

The  formula  for  determining  the  distance  of  a  planet 
from  its  periodical  revolution,  and  conversely. 

The  principle  of  rules  containing  inverted  lines  is  shown 
as  follows ; — Jjet  a,  b,  and  c,  denote  any  logarithmic 


INSTRUMENTS   AND    THE    SLIDE-RULE.-  225 

distance  on  the  A,  B,  and  C  lines ;  and,  in  lieu  of  D,  let 
an  inverted  line  A  be  laid  down,  so  that  unity  upon  it 
coincides  with  the  extremity  of  the  C  line.  Then  the 

value  of  the  same  distance  upon  this  line  will  be  -j ;  but 
if  it  be  drawn  aside  until  some  other  number  r  fall  under 

T 

the  extremity,  then  its  value  will  be  — ;  and  .-.  we  shall 

have—;  :  c  : :  a  :  Z>,  whence  b  =  - — >  where  r  is  a  constant 
A  r 

divisor,  and  A,  c,  a,  any  three  numbers. 


Solutions  of  the  more  difficult  Questions. 

Example.  161,  page  160.—  1  :  6  :  :  3  :  18,  the  depth  of 

2  /2V 

the  entire  cone  ;  hence  •$  of  the  depth  is  cut  off;  .-.  (  ^  ) 

O  N>' 

8 
or  £=•  of  the  solidity  is  cut  off,  and  the  remaining  frustum 

19     4  76  8          116 

18  07  >  5  O*  this  is  yrr^j  which  added  to  ^=  =  TTT-T  ; 

hence  135  :  18s  :  :  116  :  ?3 

135  E  :  18  D  :  :  116  E  :  17.12  D,  the  distance  from  the 
surface  of  the  water  to  the  bottom  of  the  cone  :  hence 
17.12  —  12  =  5.12,  the  depth  of  the  water. 

Ex.  162.—  2  :  18  :  :  3  :  27,  the  height  of  the  entire 


pyramid;  hence  -  of  the  height  is  cut  off;  .-.  {=J  or^y 

26 
of  the  whole  is  cut  off,  and  the  remaining  frustum  is  ~  • 

1    .  ,.  .   .   26  26       ,  3   ' 

-  oi  this  is  —  ;  so  each  person  will  have  TT=-»  and  —  will 
o  ol  ol  ol 

be  for  waste.     The  various  bulks  will   therefore  be  aa 
3,  29,  55,  and  81. 


226  A  TREATISE   ON   A  BOX  OP 

Hence  81  :  273 : :  55  :  ?3  : :  29  :  ?3  : :  3  :  93  j 

81 E :  27 D : :  55J5J :  23.73  D  : :  29  E:  19. 17  D  : :  3  E: 9  D; 

then  27  —  23.73  ==  3.27 ;  23.73  — 19.17  =  4.56 ;  19.17 

—  9=10.17. 

Ex.  172.— 4  Ib.  :  33  : :  108  Ib.  :  ?3  4  E  :  3  D  : :  108 
E :  9  inches  D,  the  diameter  of  the  globe ;  to  find  the 
content  of  which  in  gallons,  the  globe  gauge  point  is  23 ; 
divide,  then,  by  23 a,  9  times  9a. 

23  D  :  9  C  : :  9  D  :  1.37  gallons  C; 

12  +  1.37  =  13.37  gallons,  the  quantity  virtually 
put  into  the  vessel. 

Again,  5  :  20  : :  15  :  60,  the  height  of  the  entire 
pyramid ;  to  find  the  content  of  which,  in  gallons,  the 
pentagonal  pyramid  gauge  point  is  21.98 ;  divide,  then, 
by  21.983,  60  times  153. 

21.98  D  :  60  C  : :  15  D  :  27.92  gallons  (7; 

27.92  —  13.37  ==  14.55,  the  content  of  the  pyramidal 
segment  above  the  surface  of  the  water;  then  27.92  :  603 
: :  14.55  :  ?3 

27.92  E :  60  D::  14.55  E  :  48.28  inches  D,  the  height 
of  the  segmental  pyramid  above  the  water;  .-.  48.28  —  40 
=  8.28,  the  depth  of  the  vessel  unoccupied. 

1  /1\3 

Ex.  173. —  As  -  of  the  diameter  is  to  be  left,  ( •=• )  or 
5  V5/ 

1  124 

— -  of  the  solidity  will  be  left,  and  j^  will  be  turned 

down;    .-.  each  will  turn  down  — - ;  the  various  bulks 

L2o 

will  therefore  be  as  1,  32,  63,  94,  and  125. 

Hence  125  :  103  : :  94  :  ?3  : :  63  :  ?3  : :  32  :  ?3  : :  1  :  23; 

125  E :  10  D  : :  94  E  :  9.09  D  : :  63  E  :  7.96  D-..32E 
:  6.35  D::IE:2D. 


INSTRUMENTS   AND    THE    SLIDE-RULE.  '2'27 

Ex.  174.  —  To  find,  in  pints,  the  contents  of  a  globe 
whose  diameter  is  3.6  inches.  The  pint  gauge  point  for 
globes  is  8.13  ;  divide,  then,  by  8.13a,  3.6  times  3.63. 

8.13  D  :  3.6  C::  3.6  D  :  .  703(7; 

.703  +  I  =  .703  +  .777  =  1.48; 
y 

then  .703  :  3.63  :  :  1.48  :  ?3  .703  E  :  3.6  D  ::  1.48  E 
:  4.615  D,  the  diameter;  and  113  A  :  355  B  :  :  4.6154 
:  14.49.5,  the  circumference. 

Ex.  175.  —  Let  the  diameters  be  30  and  50.  The 
round  or  conic  gauge  point  for  gallons  is  46  ;  the  con- 
tent, therefore,  by  formula  17,  page  137,  is 

D 

30 

=?     viz.  50  =  14.2 

46 


55.5  gallons, 

the  content  of  a  vessel  whose  depth  is  12  inches,  and 
diameters  30  and  50.  Then,  since  the  depth  remains 
unaltered,  the  content  will  vary  as  the  squares  of  the 
diameters  ; 

hence  55.5  gals.  :  j  ^  1  :  :  14  gals.  :  1* 

55.5  C  :  30  D  :  :  14  G  :  15.06  D,  bottom  diameter; 
55.5  C  :  50  D  :  :  14  C  :  25  1  Z>,  top  diameter. 

Round  Timber. 

Instead  of  using  the  quarter  girt,  as  mentioned  at  page 
176,  it  will  be  preferable  to  take  the  tchole  girt,  andybwr 
times  the  divisor;  that  -is,  putting  L  length  in  feet,  y  girt 
in  inches,  then  the  content  by  the  common  method  will  be 

!j  thus,  in  question  183,  48  D  :  48  C  :  :  39  D  :  31.7 


228          TREATISE   ON   A   BOX   OP  INSTRUMENTS. 


feet  C.     To  find  the  true  content  the  formula  will  be 

~^-'}  thus,  in  question  184,  42.53  D  :  48  O  :  :  39  D 
4.4.  oo 

:  40  J  cubic  feet  a 

Casks. 

The  following  exhibits  the  formulae  for  the  four  varie- 
ties under  the  simplest  form  :  — 

1st  var.       £(#'  +  2.1*)       2d  var.       £(g'+2.l'—  ^of  (2diff.)« 
Fr.Pro.  Sphd.         g2  54a  Fr.  Par.  Spin.  g2  54* 


3dTar.  4th  var. 

Fr.TwoParb.  26.6a  Fr.TwoCo. 


It  will  be  found  a  great  improvement  to  the  rule  to 
copy  the  formulae  at  pp.  136,  137,  138,  on  the  back  of 
one  of  the  slides. 

June,  1848. 


THE   END. 


STEREOTYPED   BY  I.   JOHNSON  A  Ofil 


CATALOGUE 

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PUBLISHED  BT 

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A  RMENGATTD,  AMOTTBOTIX,  AND  JOHNSON.— THE  PBACTICAL 
**  DBAUGHTSMAN'S  BOOK  OF  INDTJSTBIAL  DESIGN,  AND 
MACHINIST'S  AND  ENGINEEB'S  DBAWING  COMPANION: 
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•nLENKARN.— PRACTICAL  SPECIFICATIONS  OF  WORKS  EXE- 
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HENRY  CAREY  BAIRD'S  CATALOGUE.  3 

.— MARBLE  WORKER'S  MANUAL: 
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cerning American  Marbles.  12mo.,  cloth  .  .  $1  50 

T>OOTH  AND  MORFIT.— THE  ENCYCLOPEDIA  OF  CHEMISTRY, 
a    PRACTICAL  AND  THEORETICAL  : 

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Applied  Chemistry  in  the  Franklin  Institute,  etc.,  assisted  by 
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DOWDITCH.— ANALYSIS,  TECHNICAL  VALUATION,  PURIFI- 

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•nOX.— PRACTICAL  HYDRAULICS : 

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•pOCKMASTER— THE  ELEMENTS  OF  MECHANICAL  PHYSICS  : 
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of  Alines ;  Certified  Teacher  of  Science  by  the  Department  of 
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Royal  College  of  Preceptors ;  and  late  Lecturer  in  Chemistry 
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8vo $35* 


HENRY  CAREY  BAIRD'S    CATALOGUE. 


B 


ULLOCX.  —  THE    RUDIMENTS     OF     ARCHITECTURE    AND 
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•DYR7T.— THE  COMPLETE  PRACTICAL  DISTILLER : 
»  Comprising  the  most  perfect  and  exact  Theoretical  and  Prac- 
tical Description  of  the  Art  of  Distillation  and  Rectification ; 
including  all  of  the  most  recent  improvements  in  distilling 
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With  numerous  engravings.  In  one  volume,  12mo.  $1  50 


HENRY  CAREY  BAIRD'S  CATALOGUE.  5 

pYRNE.— POCKET  BOOK  FOE  EAILEOAD   AND   CIVIL  ENGI- 

a    NEEES : 

Containing  New,  Exact,  and  Concise  Methods  for  Laying  out 
.Railroad  Curves,  Switches,  Frog  Angles  and  Crossings;  the 
Staking  out  of  work;  Levelling;  the  Calculation  of  Cut- 
tings; Embankments;  Earth-work,  etc.  By  OLIVE*  BYRXE. 
Illustrated,  ISiao.,  full  bound $1  75 

•DYENE.— THE  HANDBOOK  FOE  THE  ARTISAN,  MECHANIC, 
0    AND  ENGINEEE : 

By  OLIVER  BYRXE.     Illustrated  by  1S5  Wood  Engravings.     8vo. 

$5  00 

•D7ENE.— THE  ESSENTIAL  ELEMENTS  OF  PRACTICAL   YE- 

-0     CHANICS : 

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T)7SNE.— THE  PRACTICAL  METAL-WORKER'S  ASSISTANT: 
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Metals  and  Alloys ;  Forging  of  Iron  and  Steel ;  Hardening  and 
Tempering ;  Melting  and  Mixing ;  Casting  and  Founding ; 
Works  in  Sheet  Metal ;  the  Processes  Dependent  on  the 
Ductility  of  the  Metals ;  Soldering ;  and  the  most  Improved 
Processes  and  Tools  employed  by  Metal-Workers.  With  the 
Application  of  the  Art  of  Electro-Metallurgy  to  Manufactu- 
ring Processes ;  collected  from  Original  Sources,  and  from  the 
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others.  By  OLIVER  BYBXE.  A  New,  Revised,  and  improved 
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T)YRNE.— THE  PEACTICAL  MODEL  CALCULATOR: 

For  the  Engineer,  Mechanic,  Manufacturer  of  Engine  Work, 
Naval  Architect,  Miner,  and  Millwright.  By  OLIVER  BYRXE. 
1  volume,  8vp.,  nearly  600  pages  .  .  .  .  $4  50 

•DEMROSE.— MANUAL  OF  WOOD  CARVING :  With  Practical  II- 
lusttations  for  Learners  of  the  Art,  and  Original  and  Selected  de- 
signs. By  WILLIAM  BEMROSE,  Jr.  With  an  Introduction  by 
LLEWELLYN  JEWITT,  F.  S.  A.,  etc.  With  128  Illustrations.  4to., 
cloth $3  00 


6  HENRY  CAREY  BAIRD'S  CATALOGUE. 

•DAIRD.— PROTECTION  OF  HOME  LABOR  AND  HOME    PRO- 
•°    DUCTIONS   NECESSARY   TO   THE   PROSPERITY    OF    THE 

AMERICAN  FARMER: 

By  HENRY  CAREY  BAIRD.     8vo.,  paper      .  10 

'DAIRD.— THE  RIGHTS  OF  AMERICAN  PRODUCERS,  AND  THE 

•°    WRONGS  OF  BRITISH  FREE  TRADE  REVENUE  REFORM. 

By  HENET  CAREY  BAIRD.     (1870)  ....          5 

•DAIRD.— SOME  OF  THE  FALLACIES  OF  BRITISH-FREE-TRADE 
REVENUE-REFORM. 

Two  Letters  to  Prof.  A.  L.  Perry,  of  Williams  College,  Mass.  By 
HENRY  CAREY  BAIRD.  (1871.)  Paper  ....  5 

•DAIRD.— STANDARD  WAGES  COMPUTING  TABLES : 

An  Improvement  in  all  former  Methods  of  Computation,  so  ar- 
ranged that  wages  for  days,  hours,  or  fractions  of  hours,  at  a  spe- 
cified rate  per  day  or  hour,  may  be  ascertained  at  a  glance.  By 
T.  SPANGLER  BAIRD.  Ohlong  folio $5  00 

•DAUERMAN.— TREATISE  ON  THE  METALLURGY  OF  IRON. 
Illustrated.     12mo $2  50 

•DICKNELL'.S  VILLAGE  BUILDER. 

•^    55  large  plates.     4to $10  00 

•DISHOP.— A  HISTORY  OF  AMERICAN  MANUFACTURES : 

From  1R08  to  1866  ;  exhibiting  the  Origin  and  Growth  of  the  Prin- 
cipal Mechanic  Arts  and  Manufactures,  from  the  Earliest  Colonial 
Period  to  the  Present  Time  ;  By  J.  LEANDER  BISHOP,  M.  D.,  ED- 
WARD YOUNG,  and  EDWIN  T.  FREEDLEY.  Three  vols.  8vo., 

$10  00 

•pOX— A  PRACTICAL  TREATISE  ON  HEAT  AS  APPLIED  TO 

•°    THE  USEFUL  ARTS: 

For  the  use  of  Engineers,  Architects,  etc.  By  THOMAS  Box,  au- 
thor of  "Practical  Hydraulics."  Illustrated  by  14  plates,  con- 
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CABINET  MAKER'S  ALBUM  OF  FURNITURE  : 

Comprising  a  Collection  of  Designs  for  the  Newest  and  Most 
Elegant  Styles  of  Furniture.  Illustrated  by  Eorty-eight  Large 
and  Beautifully  Engraved  Plates.  In  one  volume,  oblong 

$5  OG 
pHAPMAN.— A  TREATISE  ON  ROPE-MAZING : 

As  practised  in  private  and  public  Rope-yards,  with  a  Description 
of  the  Manufacture,  Rules,  Tables  of  Weights,  etc.,  adapted  to  the 
Trade ;  Shipping,  Mining,  Railways,  Builders,  etc.  By  ROBERT 
CHAPMAN.  24mo •  .  .  .  $1  50 


HENRY  CAREY  BAIRD'S  CATALOGUE.  7 

pRAIK.—  THE   PRACTICAL   AMERICAN    MILLWRIGHT   AND 
^     MILLER. 

Comprising  the  Elementary  Principles  of  Mechanics,  Me- 
chanism, and  Motive  Power,  Hydraulics  and  Hydraulic 
Motors,  Mill-dams,  Saw  Mills,  Grist  Mills,  the  Oat  Meal  Mill, 
the  Barley  Mill,  Wool  Carding,  and  Cloth  Fulling  and  Dress- 
ing, Wind  Mills,  Steam  Power,  &c.  By  DAVID  CKAIK,  Mill- 
wright. Illustrated  by  numerous  wood  engravings,  and  five 
folding  plates.  1  vol.  8vo.  .  .  .  .  $5  00 


.—  A  PRACTICAL  TREATISE  ON  MECHANICAL   EN- 
GINEERING: 

Comprising  Metallurgy,  Moulding,  Casting,  Forging,  Tools, 
Workshop  Machinery,  Mechanical  Manipulation,  Manufacture 
of  Steam-engines,  etc.  etc.  With  an  Appendix  on  the  Ana- 
lysis of  Iron  and  Iron  Ores.  By  FRANCIS  CAMPIN,  C.  E.  To 
which  are  added,  Observations  on  the  Construction  of  Steam 
Boilers,  and  Remarks  upon  Furnaces  used  for  Smoke  Preven- 
tion ;  with  a  Chapter  on  Explosions.  By  R.  Armstrong,  C.  E., 
and  John  Bourne.  Rules  for  Calculating  the  Change  Wheels 
for  Screws  on  a  Turning  Lathe,  and  for  a  Wheel-cutting 
Machine.  By  J.  LA  NICCA.  Management  of  Steel,  including 
Forging,  Hardening,  Tempering,  Annealing,  Shrinking,  and 
Expansion.  And  the  Case-hardening  of  Iron.  By  G.  EDB. 
8vo.  Illustrated  with  29  plates  and  100  wood  engravings. 

$G  00 

p  \MPIN.—  THE    PRACTICE    OF  HAND-TURNING  IN  WOOD, 

U     IVORY,  SHELL,  ETC.: 

With  Instructions  for  Turning  such  works  in  Metal  as  may  be 
required  in  the  Practice  of  Turning  Wood,  Ivory,  etc.  Also 
an  Appendix  on  Ornamental  Turning.  By  FEANCIS  CAMPIN  , 
with  Numerous  Illustrations,  12mo.,  cloth  .  .  $3  00 

p  \PRON  DE  DOLE.—  DUSSAUCE.—  BLUES  AND  CARMINES  OF 
V     INDIGO. 

A  Practical-  Treatise  on  the  Fabrication  of  every  Commercial 
Product  derived  from  Indigo.  By  FELICIEN  CAPRON  DE  DOLE 
Translated,  with  important  additions,  by  Professor  II.  DCS- 
SAUCE.  12mo. 


HENRY  CAREY  BAIRD'S  CATALOGUE. 


.—  THE  WOBKS  OF  HENKY  C.  CAEEY: 

CONTRACTION  OR  EXPANSION?  REPUDIATION  OR  RE- 
SUMPTION? Letters  to  Hon.  Hugh  McCulloch.  8vo.  38 

FINANCIAL  CRISES,  their  Causes  and  Effects.     8vo.  paper 

25 

HARMONY   OF   INTERESTS;    Agricultural,   Manufacturing, 

and  Commercial.     8vo.,  paper  .....     $1  00 

Do.  do.  cloth          .         .         .     $1  50 

LETTERS  TO  THE  PRESIDENT  OF  THE  UNITED  STATES. 
Paper  .........  $1  00 

MANUAL  OF  SOCIAL  SCIENCE.  Condensed  from  Carey's 
"Principles  of  Social  Science."  By  KATE  McKEAN.  1  vol. 
12mo  ..........  $2  25 

MISCELLANEOUS  WORKS:  comprising  "Harmony  of  Inter- 
ests," "Money,"  "Letters  to  the  President,"  "French  and 
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MONEY:  A  LECTURE  before  the  N.  Y.  Geographical  and  Sta- 
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PAST,  PRESENT,  AND  FUTURE.     8vo.  .         .         .     $2  50 

PRINCIPLES  OF  SOCIAL  SCIENCE.     3  volumes  8vo.,  cloth 

$10  00 

REVIEW  OF  THE  DECADE  1857—  '67.     8vo.,  paper  50 

RECONSTRUCTION:  INDUSTRIAL,  FINANCIAL,  AND  PO- 
LITICAL. Letters  to  the  Hon.  Henry  Wilson,  U.  S.  S.  8vo, 
paper  ......  .  50 

THE  PUBLIC  DEBT,  LOCAL  AND  NATIONAL.  How  to 
provide  for  its  discharge  while  lessening  the  burden  of  Taxa- 
tion. Letter  to  David  A.  Wells,  Esq.,  U.  S.  Revenue  Commis- 
sion. 8vo.,  paper  .......  25 

THE  RESOURCES  OF  THE  UNION.  A  Lecture  read,  Dec. 
1865,  before  the  American  Geographical  and  Statistical  So- 
ciety, N.  Y.,  and  before  the  American  Association  for  the  Ad- 
vancement of  Social  Science,  Boston  .  .  .  60 

THE  SLAVE  TRADE,  DOMESTIC  AND  FOREIGN;  Why  it 
Exists,  and  How  it  may  be  Extinguished.  12mo.,  cloth  $1  5fl 


HENRY  CAREY  BAIRD'S  CATALOGUE.  9 

LETTERS    OX    INTERNATIONAL    COPYRIGHT.      (18G7.) 
Paper         .........  50 

REVIEW  OF  THE  FARMERS'  QUESTION.  (1870.)  Paper  25 
RESUMPTION!  HOW  IT  MAY  PROFITABLY  BE  BROUGHT 

AROUT.     (18C9.)    8vo.,  paper        .        .      '  .        .          60 
REVIEW  OF  THE  REPORT  OF  HON.  D.  A.  WELLS,  Special 

Commissioner  of  the  Revenue.     (18G9.)     8vo.,  paper  50 

SHALL  WE  HAVE  PEACE?   Peace  Financial  and  Peace  Poli- 

tical.   Letters  to  the  President  Elect.    (1868.)   8vo.,  paper  50 
THE    FINANCE   MINISTER   AND   THE    CURRENCY,  AND 

THE  PUBLIC  DEBT.     (1868.)     8vo.,  paper   .         .          50 
THE  WAY  TO  OUTDO  ENGLAND  WITHOUT   FIGHTING 

HER.    Letters  to  Hon.  Schuyler  Colfax.  (1865.)  8vo.,  paper 

$1  00 

WEALTH  !  OF  WHAT  DOES  IT  CONSIST  ?   (1870.)   Paper  25 
.—  A  TREATISE  OUT  THE  TEETH  OF  WHEELS: 


Demonstrating  the  best  forms  which  can  be  given  to  them  for  the 
purposes  of  Machinery,  such  as  Mill-work  and  Clock-work.  Trans- 
lated from  the  French  of  M.  CAMUS.  By  JOHN  I.  HAWKINS. 
Illustrated  by  40  plates.  Svo  ......  $3  00 


.—  MINING  LEGISLATION. 
A  paper  read  before  the  Am.  Social  Science  Association.      By 
ECKLET  B.  COXE.    Paper         ......          20 

pOLBURN.—  THE  GAS-WORKS  OF  LONDON: 

Comprising  a  sketch  of  the  Gas-works  of  the  city,  Process  of 
Manufacture,  Quantity  Produced,  Cost,  Profit,  etc.  By  ZERAH 
COLBUKX.  8\o.,  cloth  .  .....  75 

pDLBURN.—  THE  LOCOMOTIVE  ENGINE  : 

Including  a  Description  of  its  Structure,  Rules  for  Estimat- 
ing its  Capabilities,  and  Practical  Observations  on  its  Construc- 
tion and  Management.  By  ZERAH  COLBUEN.  Illustrated.  A 
new  edition.  12mo.  ......  $1  25 

pOLBURN  AND  MAW.—  THE  WATER-WORKS  OF  LONDON  : 
Together  with  a  Series  of  Articles  on  various  other  Water- 
works.    By  ZERAH  COLBURX  and  W.  MAW.     Reprinted  from 
"Engineering."     In  one  volume,  Svo.        .  .     $4  00 

r)&.GUERREOTYPIST  AND  PHOTOGRAPHER'S  COMPANION: 
^     12mo.,  cloth     ........     $1  25 


10  HENRY  CAREY  BAIRD'S  CATALOGUE. 

T\IRCKS.— PERPETUAL  MOTION : 

Or  Search  for  Self-Motive  Power  during  the  17th,  18th,  and 
19th  centuries.  Illustrated  from  various  authentic  sources  in 
Papers,  Essays,  Letters,  Paragraphs,  and  numerous  Patent 
Specifications,  with  an  Introductory  Essay  by  HENRY  DIRCKS, 
C.  E.  Illustrated  by  numerous  engravings  of  machines. 
12mo.,  cloth  . $3  50 

TlIXON.— THE  PEACTICAL  MILLWRIGHT'S  AND  ENGINEER'S 

U     GTTIDE : 

Or  Tables  for  Finding  the  Diameter  and  Power  of  Cogwheels  ; 
Diameter,  Weight,  and  Power  of  Shafts ;  Diameter  and  Strength 
of  Bolts,  etc.  etc.  By  THOMAS  DIXON.  12mo.,  cloth.  $1  50 

TpNC AN.— PRACTICAL  SURVEYOR'S  GUIDE: 

Containing  the  necessary  information  to  make  any  person,  of 
common  capacity,  a  finished  land  surveyor  without  the  aid  of 
a  teacher.  By  ANDREW  DUNCAN.  Illustrated.  12mo.,  cloth. 

$1  25 

TjUSSAUCE.— A  NEW  AND    COMPLETE    TREATISE    ON  THE 
**     ARTS  OF  TANNING,  CURRYING,  AND  LEATHER  DRESS- 
ING: 

Comprising  all  the  Discoveries  and  Improvements  made  in 
France,  Great  Britain,  and  the  United  States.  Edited  from 
Notes  and  Documents  of  Messrs.  Sallerou,  Grouvelle,  Duval, 
Dessables,  Labarraque,  Payen,  Rene",  De  Fontenelle,  Mala- 
peyre,  etc.  etc.  By  Prof.  H.  DUSSAUCE,  Chemist.  Illustrated 

by  212  wood  engravings.     8vo $10  00 

•nUSSAUCE.— A  GENERAL  TREATISE  ON  THE  MANUFACTURE 
•^    OF  SOAP,  THEORETICAL  AND  PRACTICAL : 

Comprising  the  Chemistry  of  the  Art,  a  Description  of  all  the  Raw 
Materials  and  their  Uses.  Directions  for  the  Establishment  of  a 
Soap  Factory,  with  the  necessary  Apparatus,  Instructions  in  the 
Manufacture  of  every  variety  of  Soap,  the  Assay  and  Determination 
of  the  Value  of  Alkalies,  Fatty  Substances,  Soaps,  etc.  etc.  By 
PROFESSOR  H.  DUSSAFCE.  With  an  Appendix,  containing  Ex- 
tracts from  the  Reports  of  the  International  Jury  on  Soaps,  as 
exhibited  in  the  Paris  Universal  Exposition,  18(57,  numerous 
Tables,  etc.  etc.  Illustrated  by  engravings.  In  one  volume  8vo. 

of  over  800  pages      . $10  00 

TjUSSAUCE.— PRACTICAL  TREATISE  ON  THE  FABRICATION 
".     OF  MATCHES,   GUN  COTTON,  AND  FULMINATING  POW- 
DERS. 
By  Professor  H.  DUSSAUCE.     12mo.  .         .         .     $3  00 


HENRY  CARET  BAIRD'S  CATALOGUE.  11 

TjTJSSAUCE.— A  PHACTICAL  GUIDE  FOE  THE  PERFUMER: 
Being  a  New  Treatise  on  Perfumery  the  most  favorable  to  the 
Beauty  without  being  injurious  to  the  Health,  comprising  a 
Description  of  the  substances  used  in  Perfumery,  the  Form- 
ulae of  more  than  one  thousand  Preparations,  such  as  Cosme- 
tics, Perfumed  Oils,  Tooth  Powders,  Waters,  Extracts,  Tinc- 
tures, Infusions,  Yinaigres,  Essential  Oils,  Pastels,  Creams, 
Soaps,  and  many  new  Hygienic  Products  not  hitherto  described. 
Edited  from  Notes  and  Documents  of  Messrs.  Debay,  Lunel, 

etc.    With  additions  by  Professor  H.DUSSAUCE,  Chemist.    12mo. 

$3  00 

HUSSAUCE.— A  GENERAL  TREATISE  ON  THE  MANUFACTURE 
•**    OF  VINEGAR,  THEORETICAL  AND  PRACTICAL. 

Comprising  the  various  methods,  by  the  slow  and  the  quick  pro- 
cesses, with  Alcohol,  Wine,  Grain,  Cider,  and  Molasses,  as  well 
as  the  Fabrication  of  Wood  Vinegar,  etc.  By  Prof.  H.  DUSSA.UCB. 
I2mo.  $5  00 

nUPLAIS.— A  COMPLETE  TREATISE  ON  THE  DISTILLATION 
U    AND  MANUFACTURE  OF  ALCOHOLIC  LIQUORS : 

From  the  French  of  M.  DUPLAIS.  Translated  and  Edited  by  M. 
MCKEXNIE,  M  D.  Illustrated  by  numerous  large  plates  and  wood 
engravings  of  the  best  apparatus  calculated  for  producing  the 
finest  products.  In  one  vol.  royal  8vo.  $10  00 

K7"  This  is  a  treatise  of  the  highest  scientific  merit  and  of  the 
greatest  practical  value,  surpassing  in  these  respects,  as  well  as 
in  the  variety  of  its  contents,  any  similar  volume  in  the  English 
language. 

iE  GRAFF.— THE  GEOMETRICAL  STAIR-BUILDERS'  GUIDE: 
Being  a  Plain  Practical  System  of  Hand-Railing,  embracing  all 
its  necessary  Details,  and  Geometrically  Illustrated  by  22  Steel 
Engravings  :  together  with  the  use  of  the  most  approved  princi- 
ples of  Practical  Geometry.  By  SIMOS  DK  GBAFF,  Architect. 

4to $5  °° 

T)YER  AHD  COLOR-MAKER'S  COMPANION  : 

Containing  upwards  of  two  hundred  Receipts  for  making  Co- 
lors, on  the  most  approved  principles,  for  all  the  various  styles 
and  fabrics  now  in  existence ;  with  the  Scouring  Process,  and 
plain  Directions  for  Preparing,  Washing-off,  and  Finishing  the 
Goods.  In  one  vol.  12mo. $1  25 


D 


12  HENRY  CAREY  BAIRD'S  CATALOGUE. 


pASTON.— A  PRACTICAL  TEEATISE  ON  STREET   OR  HORSE- 
*•*     POWER  RAILWAYS : 

Their  Location,  Construction,  and  Management ;  with  General 
Plans  and  Rules  for  their  Organization  and  Operation;  toge- 
ther with  Examinations  as  to  their  Comparative  Advantages 
over  the  Omnibus  System,  and  Inquiries  as  to  their  Value  for 
Investment ;  including  Copies  of  Municipal  Ordinances  relat- 
ing thereto.  By  ALEXANDER  EASTON,  C.  E.  Illustrated  by  23 
plates,  8vo.,  cloth $2  00 

pDRSYTH.— BOOK  OF  DESIGNS  FOR  HEAD-STONES,  MURAL, 
L      AND  OTHER  MONUMENTS  : 

Containing  78  Elaborate  and  Exquisite  Designs.     By   FORSYTE. 

4to.,  cloth $5  00 

*#*  This  volume,  for  the  beauty  and  variety  of  its  designs,  has 
never  been  surpassed  by  any  publication  of  the  kind,  and  should 
be  in  the  hands  of  every  marble-worker  who  does  fine  monumental 
work. 

pAIRBAIRN.— THE  PRINCIPLES  OF  MECHANISM  AND  MA- 
£      CHINERY  OF  TRANSMISSION  : 

Comprising  the  Principles  of  Mechanism,  Wheels,  and  Pulleys, 
Strength  and  Proportions  of  Shafts,  Couplings  of  Shafts,  and 
Engaging  and  Disengaging  Gear.  By  WILLIAM  FAIRBAIRN, 
Esq.,  C.  E.,  LL.  D.,  F.  R.  S.,  F.  G.  S.,  Corresponding  Member 
of  the  National  Institute  of  France,  and  of  the  Royal  Academy 
of  Turin ;  Chevalier  of  the  Legion  of  Honor,  etc.  etc.  Beau- 
tifully illustrated  by  over  150  wood-cuts.  In  one  volume  12mo. 

$2  50 

pAIRBAIRN.— PRIME-MOVERS : 

Comprising  the  Accumulation  of  Water-power ;  the  Construc- 
tion of  Water-wheels  and  Turbines;  the  Properties  of  Steam; 
the  Varieties  of  Steam-engines  and  Boilers  and  Wind-mills. 
By  WILLIAM  FAIRBAIRN,  C.  E  ,  LL.  I).,  F.  R.  S.,  F.  G.  S.  Au- 
thor of  "Principles  of  Mechanism  and  the  Machinery  of  Trans- 
mission." With  Numerous  Illustrations.  In  one  volume.  (In 
press.) 

niLBART.— A  PRACTICAL  TREATISE  ON  BANKING: 

By  JAMES  WILLIAM  GILBART.  To  which  is  added:  THE  NA- 
TIONAL BANE  ACT  AS  NOW  IN  FORCE.  8vo.  .  .  $4  50 

rtESNER.— A  PRACTICAL  TREATISE  ON  COAL,  PETROLEUM, 
U     AND  OTHER  DISTILLED  OILS. 

By  ABRAHAM  GESNER,M.  D.,  F.  G.  S.  Second  edition,  revised 
and  enlarged.  By  GEOKGE  WELTDEN  GESNER,  Consulting 
Chemist  and  Engineer.  Illustrated.  8vo.  .  .  (3  GO 


HENKY  CAREY  BAIRD'S  CATALOGUE.  13 


Q OTHIC  ALBUM  FOE  CABINET  MAKEES : 

Comprising  a  Collection  of  Designs  for  Gothic  Furniture.  Il- 
lustrated by  twenty-three  large  and  beautifully  engraved 
plates.  Oblong $3  00 

HR&NT.— BEET-EOOT    SUGAB    AND  CULTIVATION   OF  THE 
U     BEET : 

By  E.  B.  GRANT.     12mo $1  25 

rTREGOEY.— MATHEMATICS  FOE  PEACTICAL  MEN  : 

Adapted  to  the  Pursuits  of  Surveyors,  Architects,  Mechanics, 
and  Civil  Engineers.  By  OLINTHUS  GREGORY.  8vo.,  plates, 
cloth $3  00 

/VRISWOLD.— EAILEOAD  ENGINEEE'S  POCKET  COMPANION. 

Comprising  Rules  for  Calculating  Deflection  Distances  and 
Angles,  Tangential  Distances  and  Angles,  and  all  Necessary 
Tables  for  Engineers;  also  the  art  of  Levelling  from  Prelimi- 
nary Survey  to  the  Construction  of  Railroads,  intended  Ex- 
pressly for  the  Young  Engineer,  together  with  Numerous  Valu- 
able Rules  and  Examples.  By  W.  GRISWOLD.  12mo.,  tucks. 

$1  75 
nUETTIEE.— METALLIC  ALLOYS  : 

Being  a  Practical  Guide  to  their  Chemical  and  Physical  Pro- 
perties, their  Preparation,  Composition,  and  Uses.  Translated 
from  the  French  of  A.  GUETTIER,  Engineer  and  Director  of 
Founderies,  author  of  "  La  Fouderie  en  France,"  etc.  etc.  By 
A.  A.  FESQUET,  Chemist  and  Engineer.  In  one  volume,  12mo. 

$3  Of) 

TTA.TS  AND  FELTING: 

A  Practical  Treatise  on  their  Manufacture.     By  a  Practical 

Hatter.     Illustrated  by  Drawings  of  Machinery,  &c.,  8vo. 

$1  25 
TTAY.— THE  INTEEIOE  DECOEATOE : 

The  Laws  of  Harmonious  Coloring  adapted  to  Interior  Decora- 
tions :  "with  a  Practical  Treatise  on  House-Painting.  By  D. 
R.  HAY,  House-Painter  and  Decorator.  Illustrated  by  a  Dia- 
gram of  the  Primary,  Secondary,  and  Tertiary  Colors.  12mo. 

$2  25 

TTUGHES.— AMEEICAN    MILLEE    AND    MILLWEIGHT'S    AS- 

11     S  1ST  ANT  : 

I'y  W.M.  CARTER  HUGHES.  A  new  edition.  In  one  volume, 
12mo.  .  ....  $1  60 


14  HENRY  CAREY  BATRD'S  CATALOGUE. 

TTUNT.— THE  PRACTICE  OF  PHOTOGRAPHY. 

By  ROBERT  HUNT,  Vice- President  of  the  Photographic  Society, 
London.    With  numerous  illustrations.    12mo.,  cloth  .  75 


TTURST.— A  HAND-BOOK  FOE  ARCHITECTURAL  SURVEYORS : 

Comprising  Formulas  useful  in  Designing  Builders'  work,  Table 
of  Weights,  of  the  materials  used  in  Building,  Memoranda 
connected  with  Builders'  work,  Mensuration,  the  Practice  of 
Builders'  Measurement,  Contracts  of  Labor,  Valuation  of  Pro- 
perty, Summary  of  the  Practice  in  Dilapidation,  etc.  etc.  By 
J.  F.  HUEST,  C.  E.  2d  edition,  pocket-book  form,  full  bound 

$2  60 

TERVIS.— RAILWAY  PROPERTY: 

A  Treatise  on  the  Construction  and  Management  of  Railways ; 
designed  to  afford  useful  knowledge,  in  the  popular  style,  to  the 
holders  of  this  class  of  property ;  as  well  as  Railway  Mana- 
gers, Officers,  and  Agents.  By  JOHN  B.  JERVIS,  late  Chief 
Engineer  of  the  Hudson  River  Railroad,  Croton  Aqueduct,  &c. 
One  vol.  12mo.,  cloth  ....  .  $2  00 


JOHNSON.— A  REPORT  TO  THE  NAVY  DEPARTMENT  OF  THE 
0      UNITED  STATES  ON  AMERICAN  COALS : 

Applicable  to  Steam  Navigation  and  to  other  purposes.  By 
WALTER  R.  JOHNSON.  With  numerous  illustrations.  GOT  pp. 
8vo.,  .  ...  $10  00 


JOHNSTON.— INSTRUCTIONS  FOR  THE  ANALYSIS  OF  SOILS, 
U      LIMESTONES,  AND  MANURES- 

By  J.  W.  F.  JOHNSTON.     12mo 35 


T7-EENE.— A  HAND-BOOK  OF  PRACTICAL  GAUGING, 

For  the  Use  of  Beginners,  to  which  is  added  a  Chapter  on  Dis- 
tillation, describing  the  process  in  operation  at  the  Custom 
House  for  ascertaining  the  strength  of  wines.  By  JAMES  B. 
KEENE,  of  II.  M.  Customs.  8vo.  .  .  $1  25 


HENRY  CAREY  BAIRD'S  CATALOGUE.  15 

.— A  TREATISE  OH  A  BOX  OF  INSTRUMENTS, 
And  the  Slide  Rule ;  with  the  Theory  of  Trigonometry  and  Lo- 
garithms, including  Practical  Geometry,  Surveying,  Measur- 
ing of  Timber,  Cask  and  Malt  Gauging,  Heights,  and  Distances. 
By  THOMAS  KESTISH.     In  one  volume.     12mo.  .        .    $1  25 


T7-OBELL. — ERNI. — MINERALOGY 

A  thorf  method  of  Determining  and  Classifying  Minerals,  by 
mean*^  of  simple  Chemical  Experiments  in  the  Wet  Way. 
Translated  from  the  last  German  Edition  of  F.  Vox  KOBELL, 
with  an  Introduction  to  Blowpipe  Analysis  and  other  addi- 
tions. By  HEXBI  ERXI,  M.  D.,  Chief  Chemist,  Department  of 
Agriculture,  author  of  "Coal  Oil  and  Petroleum."  In  one 
volume.  12mo.  ...  .  £-50 


T  ANDRTN.— A  TREATISE  ON  STEEL : 

Comprising  its  Theory,  Metallurgy,  Properties,  Practical  Work- 
ing, and  Use.  By  M.  H.  C.  LAXDRIX,  Jr.,  Civil  Engineer. 
Translated  from  the  French,  with  Notes,  by  A.  A.  FESQCET, 
Chemist  and  Engineer.  With  an  Appendix  on  the  Bessemer 
and  the  Martin  Processes  for  Manufacturing  Steel,  from  the 
Report  of  ABBAX  S.  HEWITT,  United  States  Commissioner  to 
the  Universal  Exposition,  Paris,  1867.  12mo.  .  .  $3  00 


TARXIN.— THE  PRACTICAL  BRASS  AND  IRON  FOUNDER'S 
**    GUIDE. 

A  Concise  Treatise  on  Brass  Founding,  Moulding,  the  Metals 
and  their  Alloys,  etc.;  to  which  are  added  Recent  Improve- 
ments in  the  Manufacture  of  Iron,  Steel  by  the  Bessemer  Pro- 
cess, etc.  etc.  By  JAMES  LABKIX.  Lite  Conductor  of  the  Brass 
Foundry  Department  in  Reany,  Xeafie  &  Co.'s  Penn  Works, 
Philadelphia.  Fifth  edition,  revised,  with  extensive  Addi- 
tions. In  one  volume.  12mo $2  25 


Ib        HENRY  CAREY  BAIRD'S  CATALOGUE. 

T  EAVITT.— FACTS  ABOUT  PEAT  AS  AN  ARTICLE  OF  FUEL: 
With  Remarks  upon  its  Origin  and  Composition,  the  Localities 
in  which  it  is  found,  the  Methods  of  Preparation  and  Manu- 
facture,  and  the  various  Uses  to  which  it  is  applicable ;  toge- 
ther  with  many  other  matters  of  Practical  and  Scientific  Inte- 
rest. To  which  is  added  a  chapter  on  the  Utilization  of  Coal 
Dust  with  Peat  for  the  Production  of  an  Excellent  Fuel  at 
Moderate  Cost,  especially  adapted  for  Steam  Service.  By  II. 
T.  LEATITT.  Third  edition.  12mo.  .  .  .  .$1  75 

TERDUX,— A    PRACTICAL   TREATISE    ON    THE    MANUFAC- 

*-*     TURE  OF  WORSTEDS  AND  CARDED  YARNS : 

Translated  from  the  French  of  CHARLES  LEROUX,  Mechanical 
Engineer,  and  Superintendent  of  a  Spinning  Mill.  By  Dr,  H. 
PAINE,  and  A.  A.  FESQUET.  Illustrated  by  12  large  plates.  In 
one  volume  8vo.  .  .  .  .  .  .  .  .  $5  00 

TESLIE  (MISS).— COMPLETE  COOKERY: 

Directions  for  Cookery  in  its  Various  Branches.  By  Miss 
LESLIE.  60th  edition.  Thoroughly  revised,  with  the  addi- 
tion of  New  Receipts.  In  1  vol.  12mo.,  cloth  .  .  $1  60 

T  ESLIE  (MISS).  LADIES'  HOUSE  BOOK  : 

a  Manual  of  Domestic  Economy.  20th  revised  edition.  12mo., 
cloth .  .  $1  25 

TESLIE    (MISS).— TWO    HUNDRED    RECEIPTS    IN    FRENCH 
*-*     COOKERY. 

12mo 50 

T  IEBER.— ASSAYER'S  GUIDE; 

Or,  Practical  Directions  to  Assayers,  Miners,  and  Smelters,  for 
the  Tests  and  Assays,  by  Heat  and  by  Wet  Processes,  for  the 
Ores  of  all  the  principal  Metals,  of  Gold  and  Silver  Coins  and 
Alloys,  and  of  Coal,  etc.  By  OSCAR  M.  LIEBEB.  12mo.,  cloth 

$1  25 

T  OVE.— THE  ART  OF  DYEING,  CLEANING,  SCOURING,  AND 
*-*     FINISHING : 

On  the  most  approved  English  and  French  methods ;  being 
Practical  Instructions  in  Dyeing  Silks,  Woollens,  and  Cottons, 
Feathers,  Chips,  Straw,  etc.;  Scouring  and  Cleaning  Bed  and 
Window  Curtains,  Carpets,  Rugs,  etc.;  French  and  English 
Cleaning,  etc.  By  THOMAS  LOVE.  Second  American  Edition,  to 
which  are  added  General  Instructions  for  the  Use  of  Aniline 
Colors.  8vo.  .  5  00 


M 


M 


HENRY  CAREY  BAIRD'S  CATALOGUE.  17 

AIN  AND  BROWN.— QUESTIONS  ON  SUBJECTS  CONNECTED 
WITH  THE  MARINE  STEAM-ENGINE  : 
And  Examination  Papers;   with  Hints  for  their  Solution.      By 
THOMAS  J.  MAI.V,  Professor  of  Mathematics,  Royal  Naval  College, 
and  THOMAS  BROWN,  Chief  Engineer,  R.  N.      12mo.,  cloth   $1  50 

TM-AIN  AND  BROWN.— THE  INDICATOR  AND  DYNAMOMETER: 

With  their  Practical  Applications  to  the  Steam-Engine.  By 
THOMAS  J.  MAIN-,  M.  A.  F.  R.,  Ass't  Prof.  Royal  Naval  College, 
Portsmouth,  and  THOMAS  BROWS,  Assoc.  Inst.  C.  E.,  Chief  En- 
.  gineer,  R.  N. ,  attached  to  the  R.  N.  College.  Illustrated.  From 
the  Fourth  London  Edition.  Svo.  ...  .  $1  50 

AIN  AND  BROWN.— THE  MARINE  STEAM-ENGINE. 
By  THOMAS  J.  MAIX,  F.  R.  Ass't  S.  Mathematical  Professor  at 
Royal  Naval  College,  and  THOMAS  BROW.V,  Assoc.  Inst.  C.  E. 
Chief  Engineer,  R.  N.  Attached  to  the  Royal  Naval  College. 
Authors  of  "  Questions  Connected  with  the  Marine  Steam-En- 
gine," and  the  '•  Indicator  and  Dynamometer."  With  numerous 
Illustrations.  In  one  volume  Svo.  .  .  .  .  .  $5  00 

T^ARTIN.— SCREW-CUTTING  TABLES,  FOR  THE  USE  OF  ME- 

•UL  CHANICAL  ENGINEERS : 

Showing  the  Proper  Arrangement  of  Wheels  for  Cutting  the 
Threads  of  Screws  of  any  required  Pitch ;  with  a  Table  for 
Making  the  Universal  Gas-Pipe  Thread  and  Taps.  By  W.  A. 
MARTIN,  Engineer.  Svo. 50 

IS— A  PLAIN  TREATISE  ON  HORSE-SHOEING. 
With  Illustrations.    By  WILLIAM  MILES,  author  of  "  The  Horse's 
Foot" 

TM-OLESWORTH.— POCKET-BOOK  OF  USEFUL  FORMUUE  AND 
JU>  MEMORANDA  FOR  CIVIL  AND  MECHANICAL  EN3INEERS. 
By  GCILFOKU  L.  MOLESWORTH,  Member  of  the  Institution  of 
Civil  Engineers,  Chief  Resident  Engineer  of  the  Ceylon  Railway. 
Second  American  from  the  Tenth  London  Edition.  In  one 
volume,  full  bound  in  pocket-hook  form  .  .  .  .  $2  00 

OORE.— THE  INVENTOR'S  GUIDE: 

Patent  Office  and  Patent  Laws :  or,  a  Guide  to  Inventors,  and  a 
Book  of  Reference  for  Judges,  Lawyers,  Magistrates,  and  others. 

By  J   G.  MOORE.    12mo.,  cloth $1  Co 

TO-APIER.— A  MANUAL  OF  ELECTRO-METALLURGY: 

Including  the  Application  of  the  Art  to  Manufacturing  Processes, 
By  JAMES  NAPIER.  Fourth  American,  from  the  Fourth  London 
edition,  revised  and  enlarged.  Illustrated  by  engravings.  In 
one  volume,  Svo $2  00 


M 


18  HENRY  CAREY  BAIRD'S  CATALOGUE. 


•KT 


.—  A  SYSTEM  OF  CHEMISTRY  APPLIED  TO  DYEING  : 
Br  JAMES  NAPIER,  F.  C.  S.  A  New  and  Thoroughly  Revised 
Edition,  completely  brought  up  to  the  present  state  of  the 
Science,  including  the  Chemistry  of  Coal  Tar  Colors.  By  A.  A. 
FESQUET,  -Chemist  and  Engineer.  With  an  Appendix  on  Dyeing 
and  Calico  Printing,  as  shown  at  the  Paris  Universal  Exposition 
of  J867,  from  the  Reports  of  the  International  Jury,  etc.  Illus- 
trated. In  one  volume  8vo.,  400  pages  .  .  .  .  $5  00 

•VTEWBESY.—  GLEANINGS    FEOM    ORNAMENTAL    ART    03? 
•"    EVERY  STYLE; 

Drawn  from  Examples  in  the  British,  South  Kensington,  Indian, 
Crystal  Palace,  and  other  Museums,  the  Exhibitions  of  1851  and 
1862,  and  the  best  English  and  Foreign  works.  In  a  series  of  one 
hundred  exquisitely  drawn  Plates,  containing  many  hundred  ex- 
amples. By  ROBERT  NEWBEKY.  4to.  ....  $15  00 

•JTICHOLSON.—  A  MANUAL  OF  THE  AST  OF  BOOK-BINDING: 

Containing  full  instructions  in  the  different  Branches  of  Forward- 
ing, Gilding,  and  Finishing.  Also,  the  Art  of  Marbling  Book- 
edges  and  Paper.  By  JAMES  B.  NICHOLSON.  Illustrated.  12mo. 
cloth  ....  .....  $2  25 

jtfORRIS.—  A  HAND-BOOK  FOB  LOCOMOTIVE  ENGINEERS  AND 

-1'    MACHINISTS: 

Comprising  the  Proportions  and  Calculations  for  Constructing 
Locomotives  ;  Manner  of  Setting  Valves  ;  Tables  of  Squares, 
Cubes,  Areas,  etc.  etc.  By  SEPTIMUS  NORRIS,  Civil  and  Me- 
chanical Engineer.  New  edition.  Illustrated,  12mo.,  cloth 

$2  00 

•MTSTROM.  —  ON    TECHNOLOGICAL    EDUCATION    AND   THE 
iN    CONSTRUCTION  OF  SHIPS  AND  SCREW  PROPELLERS: 

For  Naval  and  Marine  Engineers.  By  JOHN  W.  NTSTROM,  late 
Acting  Chief  Engineer  U.  S.  N.  Second  edition,  revised  with 
additional  matter.  Illustrated  by  seven  engravings.  12mo. 

$2  50 

Q'NEILL.—  A  DICTIONARY  OF  DYEING  AND  CALICO  PRINT- 
U    ING: 

Containing  a  brief  account  of  all  the  Substances  and  Processes  in 
use  in  the  Art  of  Dyeing  and  Printing  Textile  Fabrics  :  with  Prac- 
»  tical  Receipts  and  Scientific  Information.  By  CHAHLES  O'NEILL, 
Analytical  Chemist  ;  Fellow  of  the  Chemical  Society  of  London  ; 
Member  of  the  Literary  and  Philosophical  Society  of  Manchester  ; 
Author  of  "  Chemistry  of  Calico  Printing  and  Dyeing."  To  which 
is  added  An  Essay  on  Coal  Tar  Colors  and  their  Application  To 


HENRY  CAREY  BAIRD'S  CATALOGUE.  19 

Dyeing  and  Calico  Printing.  By  A.  A.  FESQCET,  Chemist  and 
Engineer.  With  an  Appendix  on  Dyeing  and  Calico  Printing,  as 
shown  at  the  Exposition  of  1S67,  from  the  Reports  of  the  Interna. 
tional  Jury,  etc.  In  one  volume  Svo.,  491  pages  .  .  $6  00 

QSBOSN.— THE  METALLURGY  OF  ISON  AND  STEEL : 

Theoretical  and  Practical  :  In  all  its  Branches  ;  With  Special  Re- 
ference to  American  Materials  and  Processes.  By  H.  S.  OSBORN, 
LL.  D.,  Professor  of  Mining  and  Metallurgy  in  Lafayette  College, 
Easton,  Pa.  Illustrated  by  230  Engravings  on  Wood,  and  6 
Folding  Plates.  8vo.,  972  pages $10  00 

QSBOEN.— AMEEICAN  MINES  AND  MINING  : 

U    Theoretically  and  Practically  Considered.     By  Prof.  H.  S.    Os- 
BORX,  Illustrated  by  numerous  engravings.  Svo.   (In.  preparation.) 

pAINTER,  GILDES,  AND  VAENISHEE'S  COMPANION : 

Containing  Rules  and  Regulations  in  everything  relating  to  the 
Arts  of  Painting,  Gilding,  Varnishing,  and  Glass  Staining,  with 
numerous  useful  and  valuable  Receipts;  Tests  for  the  Detection 
of  Adulterations  in  Oils  and  Colors,  and  a  statement  of  the  Dis- 
eases and  Accidents  to  which  Painters,  Gilders,  and  Varnishers 
are  particularly  liable,  with  the  simplest  methods  of  Prevention 
and  Remedy.  With  Directions  for  Graining,  Marbling,  Sign  Writ- 
ing, and  Gilding  on  Glass.  To  which  are  added  COMPLETE  INSTRUC- 
TIONS FOR  COACU  PAINTING  AND  VARNISHING.  12mo.,  cloth,  $1  50 

pALLETT.— THE    MILLEE'S,    MLLLWEIGHT'S,    AND    ENGI- 

*     NEES'S  GUIDE. 

By  HENRY-  PALLETT.     Illustrated.     In  one  vol.  12mo.      .     $3  00 

pEBKINS.— GAS  AND  VENTILATION. 

Practical  Treatise  on  Gas  and  Ventilation.  With  Special  Relation 
to  Illuminating,  Heating,  and  Cooking  by  Gas."  Including  Scien- 
tific Helps  to  Engineer-students  and  others.  With  illustrated 
Diagrams.  Bv  E.  E.  PERKINS.  12mo.,  cloth  .  .  .  $1  25 

"DEEXINS  AND  STOWE.— A  NEW  GUIDE  TO  THE  SHEET-LEON 

r     AND  BOILEE  PLATE  SOLLEE: 

Containing  a  Series  of  Tables  showing  the  Weight  of  Slabs  and 
Piles  to  Produce  Boiler  Plates,  and  of  the  Weight  of  Piles  and  the 
Sizes  of  Bars  to  Produce  Sheet-iron ;  the  Thickness  of  the  Bar 
Gauge  in  Decimals ;  the  Weight  per  foot,  and  the  Thickness  on 
the  Bar  or  Wire  Gauge  of  the  fractional  parti  of  an  inch;  the 
Weight  per  sheet,  and  the  Thickness  on  the  Wire  Gauge  of  Sheet- 
iron  of  various  dimensions  to  weigh  112  Ibs.  per  bundle;  and  the 
conversion  of  Short  Weight  into  Long  Weight,  and  Long  Weight 
into  Short.  Estimated  and  collected  by  G.  H.  PERKINS  and  J.  G- 
STOWK  .  .  ....  $259 


20  HENRY  CAREY  BAIRD'S  CATALOGUE. 

pHILLIPS  AND  DARLINGTON.—  RECORDS  OF  MINING  AND 

*     METALLURGY : 

Or,  Facts  and  Memoranda  for  the  use  of  the  Mine  Agent  and 
Smelter.  By  J.  ARTHUR  PHILLIPS,  Mining  Engineer,  Graduate  of 
the  Imperial  School  of  Mines,  France,  etc.,  and  JOIIN  DARLINGTON. 
Illustrated  by  numerous  engravings.  In  one  vol.  12mo.  .  $2  00 

pRADAL,    MALEPEYRE,     AND    DUSSAUCE.  — A    COMPLETE 

•*•     TREATISE  ON  PERFUMERY: 

Containing  notices  of  the  Raw  Material  used  in  the  Ait,  and  the 
Best  Formulae.  According  to  the  most  approved  Methods  followed 
in  France,  England,  and  the  United  States.  By  M.  P.  PRADAL, 
Perfumer-Chemist,  and  M.  F.  MALEPEYRE.  Translated  from  the 
French,  with  extensive  additions,  by  Prof.  H.  DUSSAUCE.  8vo.  $10 

pROTEAUX.— PRACTICAL   GUIDE  FOR  THE  MANUFACTURE 

*     OF  PAPER  AND  BOARDS. 

By  A.  PROTEAUX,  Civil  Engineer,  and  Graduate  of  the  School  of 
Arts  and  Manufactures,  Director  of  Thiers's  Paper  Mill,  'Puy-de- 
Dome.  With  additions,  by  L.  S.  LE  NORMAND.  Translated  from 
the  French,  with  Notes,  by  HORATIO  PAINE,  A.  B.,  M.  D.  To 
which  is  added  a  Chapter  on  the  Manufacture  of  Paper  from  Wood 
in  the  United  States,  by  HENRY  T.  BROWN,  of  the  "American 
Artisan."  Illustrated  by  six  plates,  containing  Drawings  of  Raw 
Materials,  Machinery,  Plans  of  Paper-Mills,  etc.  etc.  8vo.  $5  00 

•DEGNAULT.— ELEMENTS  OF  CHEMISTRY. 

By  M.  V.  REGNAULT.  Translated  from  the  French  by  T.  FOR- 
REST BENTON,  M.  L.,  and  edited,  with  notes,  by  JAMES  C.  BOOTH, 
Melter  and  Refiner  U.  S.  Mint,  and  WM.  L.  FABER,  Metallurgist 
and  Mining  Engineer.  Illustrated  by  nearly  700  wood  engravings. 
Comprising  nearly  1500  pages.  In  two  vols.  8vo.,  cloth  $10  00 

TDEID.— A  PRACTICAL  TREATISE  ON  THE  MANUFACIURE  OF 

•"    PORTLAND  CEMENT: 

By  HENRY  REID,  C.  E.  To  which  is  added  a  Translation  of  M. 
A.  Lipowitz's  Work,  describing  anew  method  adopted  in  Germany 
of  Manufacturing  that  Cement.  By  W.  F.  REID.  Illustrated  by 
plates  and  wood  engravings.  8vo.  .  .  .  .  .  $7  00 

TDIFFAULT,    VERGNAUD,    AND    TOUSSAINT.— A    PRACTICAL 
11   TREATISE    ON   THE    MANUFACTURE    OF    COLORS    FOR 
PAINTING: 

Containing  the  best  Formulae  and  the  Processes  the  Newest  and 
in  most  General  Use.  By  MM.  RIFFAULT,  VERGXAUD,  and  TOUS- 
SAINT. Revised  and  Edited  by  M.  F.  MALEPEYRE  and  Dr.  E>rn, 
WINCKLER.  Illustrated  by  Engravings.  In  one  vol.  8vo.  (In 
1  ffepa  ration.) 


HENRY  CAREY  BAIRD'S  CATALOGUE.  21 


TDIFFAULF,    VERSNAUD,    AND    TOUSSAINT.—  A    PRACTICAL 
TREATISE  ON  THE  MANUFACTURE  OF  VARNISHES: 
By  MM.  RIFFACLT,  VERGXAUD,  and  TOUSSAIXT.     Revised  and 
Edited  by  M.  F.  MALEPEYRE  and  Dr.  EMIL  WINCKLER.     Illus- 
trated.    In  one  vol.  8vo.     (In  preparation.) 

gHUNK.—  A  PRACTICAL  TREATISE   ON   RAILWAY   CUEVES 
°    AND  LOCATION,  FOR  YOUNG  ENGINEERS. 

By  WH.  F.  SHTJSK,  Civil  Engineer.    12mo.,  tucks    .         .     $2  00 

OMEATON.—  BUILDEE'S  POCKET  COMPANION: 

Containing  the  Elements  of  Building,  Surveying,  and  Architec. 
ture  ;  with  Practical  Rules  and  Instructions  connected  with  the  suh- 
ject.  By  A.  C.  SMEATOX,  Civil  Engineer,  etc.  In  one  volume, 
12mo  ........  ..  .  .  $1  50 


—  THE  DYER'S  INSTEUCTOR: 

Comprising  Practical  Instructions  in  the  Art  of  Dyeing  Silk,  Cot- 
ton, Wool,  and  Worsted,  and  Woollen  Goods:  containing  nearly 
800  Receipts.  To  which  is  added  a  Treatise  on  the  Art  of  Pad- 
ding; and  the  Printing  of  Silk  Warps,  Skeins,  and  Handkerchiefs, 
and  the  various  Mordants  and  Colors  for  the  different  styles  of 
such  work.  By  DAVID  SMITH,  Pattern  Dyer,  12mo.,  cloth 

$3  00 
OMITH.—  THE  PRACTICAL  DYEE'S  GUIDE: 

Comprising  Practical  Instructions  in  the  Dyeing  of  Shot  Cobourgs, 
Silk  Striped  Orleans,  Colored  Orleans  from  Black  Warps,  ditto 
from  White  Warps,  Colored  Cobourgs  from  White  Warps,  Merinos, 
Yarns,  Woollen  Cloths,  etc.  Containing  nearly  300  Receipts,  to 
most  of  which  a  Dyed  Pattern  is  annexed.  Also,  a  Treatise  on 
the  Art  of  Padding.  By  DAVID  SMITH.  In  one  vol.  8vo.  $25  00 

QIHAW.—  CIVIL  AECHITECTUEE  : 

Being  a  Complete  Theoretical  and  Practical  System  of  Building, 
containing  the  Fundamental  Principles  of  the  Art.  By  EDTV  vnn 
SHAW,  Architect.  To  which  is  added  a  Treatise  on  Gothic  Ar  hi- 
tectnre,  Ac.  By  THOMAS  W.  SILLOWAY  and  GEORGE  M.  HARD- 
ING ,  Architects.  The  whole  illustrated  by  102  quarto  plates  finely 
engraved  on  copper.  Eleventh  Edition.  4to.  Cloth.  $10  00 

OLOAN—  AMEEICAN  HOUSES: 

A  variety  of  Original  Designs  for  Rural  Buildings.  Illustrated  by 
26  colored  Engravings,  with  Descriptive  References.  By  SAMUEL 
SLOAN,  Architect,  author  of  the  "  Model  Architect,"  etc.  etc.  8vo. 

$2  50 

OCHINZ.—  RESEAECHES   ON   THE   ACTION   OF   THE   BLAST. 
FUENACE. 
By  CHAS.  SCHINZ.     Seven  plates.     12mo.          .         .        .     $4  25 


22  HENRY  CAREY  BAIRD'S  CATALOGUE. 

OlMITH.— PARKS  AND  PLEASURE  GROUNDS : 

Or,  Practical  Notes  on  Country  Residences,  Villas,  Public  Parks, 
and  Gardens.  By  CHARLES  H.  J.  SMITH,  Landscape  Gardener 
and  Garden  Architect,  etc.  etc.  12mo.  .  .  .  .  $2  25 

STOKES.— CABINET-MAKER'S  AND  UPHOLSTERER'S  COMPA- 
°    NION: 

Comprising  the  Rudiments  and  Principles  of  Cabinet-making  and 
Upholstery,  with  Familiar  Instructions,  Illustrated  by  Examples 
for  attaining  a  Proficiency  in  the  Art  of  Drawing,  as  applicable 
to  Cabinet-work  ;  The  Processes  of  Veneering,  Inlaying,  and 
Buhl- work  ;  the  Art  of  Dyeing  and  Staining  Wood,  Bone,  Tortoise 
Shell,  etc.  Directions  for  Lackering,  Japanning,  and  Varnishing  ; 
to  make  French  Polish ;  to  prepare  the  Best  Glues,  Cements,  and 
Compositions,  and  a  number  of  Receipts,  particularly  for  workmen 
generally.  By  J.  STOKES.  In  one  vol.  12mo.  With  illustrations 

$1  25 

STRENGTH  AND  OTHER  PROPERTIES  OF  METALS. 

Reports  of  Experiments  on  the  Strength  and  other  Properties  of 
Metals  for  Cannon.  With  a  Description  of  the  Machines  for  Test- 
ing Metals,  and  of  the  Classification  of  Cannon  in  service.  By 
Officers  of  the  Ordnance  Department  U.  S.  Army.  By  authority 
of  the  Secretary  of  War.  Illustrated  by  25  large  steel  plates.  In 
1  vol.  quarto .  $10  00 

OULLIVAN.— PROTECTION  TO  NATIVE  INDUSTRY. 

By  Sir  EDWARD  SULLIVAN,  Baronet.    (1870.)     8vo.        .     $1  50 

mABLES  SHOWING  THE  WEIGHT  OF  ROUND,  SQUARE,  AND 
A    FLAT  BAR  IRON,  STEEL,  ETC. 

By  Measurement.     Cloth  ......  63 

rPAYLOR.— STATISTICS  OF  COAL: 

Including  Mineral  Bituminous  Substances  employe!  in  Arts  and 
Manufactures  ;  with  their  Geographical,  Geological,  and  Commer- 
cial Distribution  and  amount  of  Production  and  Consumption  on 
the  American  Continent.  With  Incidental  Statistics  of  the  Iron 
Manufacture.  By  R.  C.  TAYLOR.  Second  edition,  revised  by  S. 
S.  HALDEMAN.  Illustrated  by  five  Maps  and  many  wood  engrav- 
ings. 8vo.,  cloth $6  00 

rpEMPLETON.— THE    PRACTICAL   EXAMINATOR    ON    STEAM 

*     AND  THE  STEAM-ENGINE  : 

With  Instructive  References  relative  thereto,  for  the  Use  of  Engi- 
neers, Students,  and  others.  By  WM.  TEMPLETON,  Engineer  ]2mo. 

$1  25 


HENRY  CARET  BAIRD'S  CATALOGUE.  23 

rpHOMAS.— THE  MODERN  PRACTICE  OF  PHOTOGRAPHY. 

•*•     By  R.  W.  THOMAS,  F.  C.  S.     8vo.,  cloth  .         ...          75 

fPHOMSON.— FREIGHT  CHARGES  CALCULATOR. 

By  ANDREW  THOMSON,  Freight  Agent  .  .  .  .  $1  25 
JIURNING :  SPECIMENS  OF  FANCY  TURNING  EXECUTED  ON 
•*•  THE  HAND  OR  FOOT  LATHE : 

With  Geometric,  Oval,  and  Eccentric  Chucks,  and  Elliptical  Cut- 
ting Frame.  By  an  Amateur.  Illustrated  by  30  exquisite  Pho- 
tographs. 4to $3  00 

BURNER'S  (THE)  COMPANION: 

Containing  Instructions  in  Concentric,  Elliptic,  and  Eccentric 
Turning;  also  various  Plates  of  Chucks,  Tools,  and  Instru- 
ments ;  and  Directions  for  using  the  Eccentric  Cutter,  Drill, 
Vertical  Cutter,  and  Circular  Rest;  with  Patterns  and  Instruc- 
tions for  working  them.  A  new  edition  in  1  vol.  ]2mo.  $1  50 

TTRBIN  — BRULL.  — A   PRACTICAL    GUIDE   FOR   PUDDLING 

U    IRON  AND  STEEL. 

By  ED.  URBIN,  Engineer  of  Arts  and  Manufactures.  A  Prize 
Essny  read  before  the  Association  of  Engineers,  Graduate  of  the 
School  of  Mines,  of  Liege,  Belgium,  at  the  Meeting  of  1865-6. 
To  which  is  added  a  COMPARISON  OF  THE  RESISTING  PROPERTIES 
OF  IRON  AND  STEEL.  By  A.  BRULL.  Translated  from  the  French 
by  A.  A.  FESQUET,  Chemist  and  Engineer.  In  one  volume,  8vo. 

$1  00 

T70GDES.— THE  ARCHITECT'S  AND  BUILDER'S  POCKE1  COM- 
V    PANION  AND  PRICE  BOOK. 

By  F.  W.  VOGDES,  Architect.  Illustrated.  Full  bound  in  pocket- 
book  form $2  00 

In  book  form,  ISmo.,  muslin     .         .         •         .         .         .       1  50 

TTtTARN.— THE  SHEEI  METAL  WORKER'S  INSTRUCTOR,  FOR 
V  ZINC,    SHEET-IRON,   COPPER   AND   TIN   PLATE   WORK- 
ERS, &c. 

By  REUBEN  HENRY  WARN,  Practical  Tin  Plate  "Worker.  I'!us- 
trated  by  32  plates  and  37  wood  engravings.  8vo.  .  $3  CO 

yn-ATSON.— A  MANUAL  OF  THE  HAND-LATHE. 

By  EGBERT  P.  WATSON,  Late  of  the  "  Scientific  American,"  Au- 
thor of  "Modern  Practice  of  American  Machinists  and  Engi- 
neers," In  one  volume,  12ino. $1  50 


24  HENRY  CAREY  BAIRD'S  CATALOGUE. 


.—  THE   MODERN   PEACTICE    OF  AMERICAN    MA- 
*      CHINISTS  AND  ENGINEERS  : 

Including  the  Construction,  Application,  and  Use  of  Drills,  Lathe 
Tools,  Cutters  for  Boring  Cylinders,  and  Hollow  Work  Generally, 
with  the  most  Economical  Speed  of  the  same,  the  Results  verified 
by  Actual  Practice  at  the  Lathe,  the  Vice,  and  on  the  Floor. 
Together  with  Workshop  management,  Economy  of  Manufacture, 
the  Steam-Engine,  Boilers,  Gears,  Belting,  etc.  etc.     By  EGBERT 
P.  WATSON,  late  of  the  "Scientific  American."     Illustrated  by 
eighty-six  engravings.     12mo.  .....     $2  50 

Tin-ATSON.—  THE  THEORY  AND  PRACTICE   OF  THE  ART  OF 
VV  WEAVING  BY  HAND  AND  POWER: 

With  Calculations  and  Tables  for  the  use  of  those  connected  •with 
the  Trade.  By  JOHN  WATSON,  Manufacturer  and  Practical  Machine 
Maker.  Illustrated  by  large  drawings  of  the  best  Power-Looms. 
8vo  ...........  $10  00 

TTTTEATHERLY.—  TREATISE    ON  .THE  ART   OF  BOILING   SU- 
""  GAR,    CRYSTALLIZING,     LOZENGE-MAKING,     COMFITS, 
GUM  GOODS, 

And  other  processes  for  Confectionery,  Ac.  In  which  are  ex- 
plained, in  an  easy  and  familiar  manner,  the  various  Methods 
of  Manufacturing  every  description  of  Raw  and  Refined  Sugar 
Goods,  as  sold  by  Confectioners  and  others  .  .  .  $2  00 

XL.—  TABLES  FOR  QUALITATIVE  CHEMICAL  ANALYSIS. 
By  Prof.  HEINRICH  WILL,  of  Giessen,  Germany.  Seventh  edi- 
tion. Translated  by  CHARLES  F.  HIMES,  Ph.  D.,  Professor  of 
Natural  Science,  Dickinson  College,  Carlisle,  Pa.  .  .  $1  25 

-TTrriLLIAMS.—  ON  HEAT  AND  STEAM  : 

Embracing  New  Views  of  Vaporization,  Condensation,  and  Expan- 
sion. By  CHARLES  WYE  WILLIAMS,  A.  I.  C.  E.  Illustrated.  8vo. 

$3  50 

•yyORSSAM.—  ON  MECHANICAL  SAWS: 

From  the  Transactions  of  the  Society  of  Engineers,  1867.  By 
6.  W.  WORSSAM,  Jr.  Illustrated  by  18  large  folding  plates.  8vo. 

$5  00 

WOHLER.—  A  HAND-BOOK  OF  MINERAL  ANALYSIS. 

"  By  F.  WOHLKR.  Edited  by  II.  B.  NASON,  Professor  of  Chemistry, 
Rensselaer  Institute,  Troy,  N.  Y.  With  numerous  Illustrations. 
12mo.  ......  $3  00 


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